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119-402: Euler both developed the techniques of analysis and applied them to numerous problems in mechanics , notably in later publications the calculus of variations . Euler's laws of motion expressed scientific laws of Galileo and Newton in terms of points in reference frames and coordinate systems making them useful for calculation when the statement of a problem or example is slightly changed from

238-497: A 0 b k + a 1 b k − 1 + ⋯ + a k − 1 b 1 + a k b 0 . {\displaystyle \!a_{0}b_{k}+a_{1}b_{k-1}+\cdots +a_{k-1}b_{1}+a_{k}b_{0}.} Here, the convergence of the partial sums of the series c 0 + c 1 + c 2 + ⋯ {\displaystyle c_{0}+c_{1}+c_{2}+\cdots }

357-583: A 0 + a 1 + a 2 + ⋯ or a 1 + a 2 + a 3 + ⋯ , {\displaystyle a_{0}+a_{1}+a_{2}+\cdots \quad {\text{or}}\quad a_{1}+a_{2}+a_{3}+\cdots ,} where the terms a k {\displaystyle a_{k}} are the members of a sequence of numbers , functions , or anything else that can be added . A series may also be represented with capital-sigma notation : ∑ k = 0 ∞

476-461: A 0 + a 1 + a 2 + ⋯ + a n + ⋯  or  f ( 0 ) + f ( 1 ) + f ( 2 ) + ⋯ + f ( n ) + ⋯ . {\displaystyle a_{0}+a_{1}+a_{2}+\cdots +a_{n}+\cdots \quad {\text{ or }}\quad f(0)+f(1)+f(2)+\cdots +f(n)+\cdots .} For example, Euler's number can be defined with

595-401: A 0 + a 2 + a 1 = {\displaystyle a_{0}+a_{2}+a_{1}={}} a 2 + a 1 + a 0 . {\displaystyle a_{2}+a_{1}+a_{0}.} Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change

714-433: A 1 + a 2 + a 3 + ⋯ , {\displaystyle a_{1}+a_{2}+a_{3}+\cdots ,} or, using capital-sigma summation notation , ∑ i = 1 ∞ a i . {\displaystyle \sum _{i=1}^{\infty }a_{i}.} The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in

833-593: A 1 + a 2 + ⋯ {\displaystyle a_{0}+a_{1}+a_{2}+\cdots } may not equal the sum of a 0 + ( a 1 + a 2 ) + {\displaystyle a_{0}+(a_{1}+a_{2})+{}} ( a 3 + a 4 ) + ⋯ . {\displaystyle (a_{3}+a_{4})+\cdots .} For example, Grandi's series ⁠ 1 − 1 + 1 − 1 + ⋯ {\displaystyle 1-1+1-1+\cdots } ⁠ has

952-435: A 1 + a 2 = {\displaystyle a_{0}+a_{1}+a_{2}={}} a 0 + ( a 1 + a 2 ) = {\displaystyle a_{0}+(a_{1}+a_{2})={}} ( a 0 + a 1 ) + a 2 . {\displaystyle (a_{0}+a_{1})+a_{2}.} Similarly, in a series, any finite groupings of terms of

1071-477: A 1 , a 2 , a 3 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} of terms, whether those terms are numbers, functions , matrices , or anything else that can be added, defines a series, which is the addition of the a i one after the other. To emphasize that there are an infinite number of terms, series are often also called infinite series . Series are represented by an expression like

1190-411: A 3 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} is summable , and otherwise, when the limit does not exist, the series is divergent . The expression ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}} denotes both the series—the implicit process of adding the terms one after

1309-397: A i = lim n → ∞ ∑ i = 1 n a i , {\displaystyle \sum _{i=1}^{\infty }a_{i}=\lim _{n\to \infty }\,\sum _{i=1}^{n}a_{i},} if it exists. When the limit exists, the series is convergent or summable and also the sequence ( a 1 , a 2 ,

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1428-470: A j b k − j , {\displaystyle {\biggl (}\sum _{k=0}^{\infty }a_{k}{\biggr )}\cdot {\biggl (}\sum _{k=0}^{\infty }b_{k}{\biggr )}=\sum _{k=0}^{\infty }c_{k}=\sum _{k=0}^{\infty }\sum _{j=0}^{k}a_{j}b_{k-j},} with each c k = ∑ j = 0 k a j b k − j = {\textstyle c_{k}=\sum _{j=0}^{k}a_{j}b_{k-j}={}\!}

1547-410: A k or ∑ k = 1 ∞ a k . {\displaystyle \sum _{k=0}^{\infty }a_{k}\qquad {\text{or}}\qquad \sum _{k=1}^{\infty }a_{k}.} It is also common to express series using a few first terms, an ellipsis, a general term, and then a final ellipsis, the general term being an expression of the n th term as a function of n :

1666-394: A k + b k . {\displaystyle \sum _{k=0}^{\infty }a_{k}+\sum _{k=0}^{\infty }b_{k}=\sum _{k=0}^{\infty }a_{k}+b_{k}.} Using the symbols s a , n {\displaystyle s_{a,n}} and s b , n {\displaystyle s_{b,n}} for the partial sums of the added series and s

1785-404: A k = lim n → ∞ s n . {\displaystyle \sum _{k=0}^{\infty }a_{k}=\lim _{n\to \infty }\sum _{k=0}^{n}a_{k}=\lim _{n\to \infty }s_{n}.} A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms. When the sum exists,

1904-433: A n {\textstyle \sum a_{n}} also fails to converge absolutely, although it could still be conditionally convergent, for example, if the a n {\displaystyle a_{n}} alternate in sign. Second is the general limit comparison test : If ∑ b n {\textstyle \sum b_{n}} is an absolutely convergent series such that |

2023-491: A n {\textstyle \sum a_{n}} converges absolutely as well. If ∑ | b n | {\textstyle \sum \left\vert b_{n}\right\vert } diverges, and | a n | ≥ | b n | {\displaystyle \left\vert a_{n}\right\vert \geq \left\vert b_{n}\right\vert } for all sufficiently large n {\displaystyle n} , then ∑

2142-432: A n {\textstyle \sum a_{n}} converges absolutely. Alternatively, using comparisons to series representations of integrals specifically, one derives the integral test : if f ( x ) {\displaystyle f(x)} is a positive monotone decreasing function defined on the interval [ 1 , ∞ ) {\displaystyle [1,\infty )} then for

2261-539: A n {\textstyle \sum a_{n}} , If ∑ b n {\textstyle \sum b_{n}} is an absolutely convergent series such that | a n | ≤ C | b n | {\displaystyle \left\vert a_{n}\right\vert \leq C\left\vert b_{n}\right\vert } for some positive real number C {\displaystyle C} and for sufficiently large n {\displaystyle n} , then ∑

2380-851: A n > 0 {\displaystyle a_{n}>0} is called alternating . Such a series converges if the non-negative sequence a n {\displaystyle a_{n}} is monotone decreasing and converges to  0 {\displaystyle 0} . The converse is in general not true. A famous example of an application of this test is the alternating harmonic series ∑ n = 1 ∞ ( − 1 ) n + 1 n = 1 − 1 2 + 1 3 − 1 4 + 1 5 − ⋯ , {\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,} which

2499-591: A n + 1 a n | ≤ | b n + 1 b n | {\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert \leq \left\vert {\tfrac {b_{n+1}}{b_{n}}}\right\vert } for sufficiently large n {\displaystyle n} , then ∑ a n {\textstyle \sum a_{n}} converges absolutely as well. If ∑ | b n | {\textstyle \sum \left|b_{n}\right|} diverges, and |

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2618-516: A n + 1 a n | ≥ | b n + 1 b n | {\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert \geq \left\vert {\tfrac {b_{n+1}}{b_{n}}}\right\vert } for all sufficiently large n {\displaystyle n} , then ∑ a n {\textstyle \sum a_{n}} also fails to converge absolutely, though it could still be conditionally convergent if

2737-428: A n + 1 a n | < C {\displaystyle \left\vert {\tfrac {a_{n+1}}{a_{n}}}\right\vert <C} for all sufficiently large  n {\displaystyle n} , then ∑ a n {\textstyle \sum a_{n}} converges absolutely. When the ratio is less than 1 {\displaystyle 1} , but not less than

2856-431: A r k = a + a r + a r 2 + ⋯ + a r n = a 1 − r n + 1 1 − r . {\displaystyle s_{n}=\sum _{k=0}^{n}ar^{k}=a+ar+ar^{2}+\cdots +ar^{n}=a{\frac {1-r^{n+1}}{1-r}}.} Strictly speaking, a series is said to converge , to be convergent , or to be summable when

2975-468: A + ( a + d ) + ( a + 2 d ) + ⋯ + ( a + n d ) = ( n + 1 ) a + 1 2 n ( n + 1 ) d , {\displaystyle s_{n}=\sum _{k=0}^{n}\left(a+kd\right)=a+(a+d)+(a+2d)+\cdots +(a+nd)=(n+1)a+{\tfrac {1}{2}}n(n+1)d,} and a geometric series has partial sums s n = ∑ k = 0 n

3094-403: A + b , n {\displaystyle s_{a+b,n}} for the partial sums of the resulting series, this definition implies the partial sums of the resulting series follow s a + b , n = s a , n + s b , n . {\displaystyle s_{a+b,n}=s_{a,n}+s_{b,n}.} Then the sum of the resulting series, i.e.,

3213-426: A , n {\displaystyle s_{ca,n}=cs_{a,n}} for all n , {\displaystyle n,} and therefore also lim n → ∞ s c a , n = c lim n → ∞ s a , n , {\textstyle \lim _{n\rightarrow \infty }s_{ca,n}=c\lim _{n\rightarrow \infty }s_{a,n},} when

3332-399: A consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure . This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ {\displaystyle \sigma } -algebra . This means that

3451-572: A limit during the 17th century, especially through the early calculus of Isaac Newton . The resolution was made more rigorous and further improved in the 19th century through the work of Carl Friedrich Gauss and Augustin-Louis Cauchy , among others, answering questions about which of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute convergence and conditional convergence of series. In modern terminology, any ordered infinite sequence (

3570-460: A set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space , which assigns the conventional length , area , and volume of Euclidean geometry to suitable subsets of

3689-474: A telescoping sum argument implies that the partial sums of the series of those non-negative bounding terms are themselves bounded above by 2. The exact value of this series is 1 6 π 2 {\textstyle {\frac {1}{6}}\pi ^{2}} ; see Basel problem . This type of bounding strategy is the basis for general series comparison tests. First is the general direct comparison test : For any series ∑

Mechanica - Misplaced Pages Continue

3808-541: A compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. Techniques from analysis are used in many areas of mathematics, including: Series (mathematics) In mathematics , a series is, roughly speaking, an addition of infinitely many terms , one after

3927-537: A constant less than 1 {\displaystyle 1} , convergence is possible but this test does not establish it. Second is the root test : if there exists a constant C < 1 {\displaystyle C<1} such that | a n | 1 / n ≤ C {\displaystyle \textstyle \left\vert a_{n}\right\vert ^{1/n}\leq C} for all sufficiently large  n {\displaystyle n} , then ∑

4046-405: A different result. In general, grouping the terms of a series creates a new series with a sequence of partial sums that is a subsequence of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then

4165-466: A finite amount of time. However, if the terms and their finite sums belong to a set that has limits , it may be possible to assign a value to a series, called the sum of the series . This value is the limit as n tends to infinity of the finite sums of the n first terms of the series if the limit exists. These finite sums are called the partial sums of the series. Using summation notation, ∑ i = 1 ∞

4284-428: A measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X {\displaystyle X} . It must assign 0 to the empty set and be ( countably ) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate

4403-410: A metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance). Formally, a metric space is an ordered pair ( M , d ) {\displaystyle (M,d)} where M {\displaystyle M} is a set and d {\displaystyle d}

4522-427: A sequence is convergence . Informally, a sequence converges if it has a limit . Continuing informally, a ( singly-infinite ) sequence has a limit if it approaches some point x , called the limit, as n becomes very large. That is, for an abstract sequence ( a n ) (with n running from 1 to infinity understood) the distance between a n and x approaches 0 as n → ∞, denoted Real analysis (traditionally,

4641-691: A sequence of partial sums that alternates back and forth between ⁠ 1 {\displaystyle 1} ⁠ and ⁠ 0 {\displaystyle 0} ⁠ and does not converge. Grouping its elements in pairs creates the series ( 1 − 1 ) + ( 1 − 1 ) + ( 1 − 1 ) + ⋯ = {\displaystyle (1-1)+(1-1)+(1-1)+\cdots ={}} 0 + 0 + 0 + ⋯ , {\displaystyle 0+0+0+\cdots ,} which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after

4760-462: A series a 0 + a 1 + a 2 + ⋯ {\displaystyle a_{0}+a_{1}+a_{2}+\cdots } is unconditionally convergent if and only if the series summing the absolute values of its terms, | a 0 | + | a 1 | + | a 2 | + ⋯ , {\displaystyle |a_{0}|+|a_{1}|+|a_{2}|+\cdots ,}

4879-421: A series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges. For series of real numbers or complex numbers, series addition is associative , commutative , and invertible . Therefore series addition gives the sets of convergent series of real numbers or complex numbers the structure of an abelian group and also gives

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4998-421: A series with terms a n = f ( n ) {\displaystyle a_{n}=f(n)} for all  n {\displaystyle n} , ∑ a n {\textstyle \sum a_{n}} converges if and only if the integral ∫ 1 ∞ f ( x ) d x {\textstyle \int _{1}^{\infty }f(x)\,dx}

5117-403: A series's sequence of partial sums is not easily calculated and evaluated for convergence directly, convergence tests can be used to prove that the series converges or diverges. In ordinary finite summations , terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of the associativity of addition. a 0 +

5236-467: A sum of the natural logarithm of 2 , while the sum of the absolute values of the terms is the harmonic series , ∑ n = 1 ∞ 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ , {\displaystyle \sum \limits _{n=1}^{\infty }{1 \over n}=1+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots ,} which diverges per

5355-673: A third series c 0 + c 1 + c 2 + ⋯ {\displaystyle c_{0}+c_{1}+c_{2}+\cdots } , called the Cauchy product, can be written in summation notation ( ∑ k = 0 ∞ a k ) ⋅ ( ∑ k = 0 ∞ b k ) = ∑ k = 0 ∞ c k = ∑ k = 0 ∞ ∑ j = 0 k

5474-667: Is 1 2 {\displaystyle {\tfrac {1}{2}}} times the original series, so it would have a sum of half of the natural logarithm of 2. By the Riemann series theorem, rearrangements of the alternating harmonic series to yield any other real number are also possible. The addition of two series a 0 + a 1 + a 2 + ⋯ {\textstyle a_{0}+a_{1}+a_{2}+\cdots } and b 0 + b 1 + b 2 + ⋯ {\textstyle b_{0}+b_{1}+b_{2}+\cdots }

5593-441: Is s n = ∑ k = 0 n a k = a 0 + a 1 + ⋯ + a n . {\displaystyle s_{n}=\sum _{k=0}^{n}a_{k}=a_{0}+a_{1}+\cdots +a_{n}.} Some authors directly identify a series with its sequence of partial sums. Either the sequence of partial sums or the sequence of terms completely characterizes

5712-459: Is ∑ k = 0 n 1 2 k = 2 − 1 2 n . {\displaystyle \sum _{k=0}^{n}{\frac {1}{2^{k}}}=2-{\frac {1}{2^{n}}}.} As one has lim n → ∞ ( 2 − 1 2 n ) = 2 , {\displaystyle \lim _{n\to \infty }\left(2-{\frac {1}{2^{n}}}\right)=2,}

5831-561: Is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders . Differential equations play a prominent role in engineering , physics , economics , biology , and other disciplines. Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives)

5950-537: Is a metric on M {\displaystyle M} , i.e., a function such that for any x , y , z ∈ M {\displaystyle x,y,z\in M} , the following holds: By taking the third property and letting z = x {\displaystyle z=x} , it can be shown that d ( x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0}     ( non-negative ). A sequence

6069-512: Is a branch of mathematical analysis dealing with vector-valued functions . Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have. Techniques from analysis are also found in other areas such as: The vast majority of classical mechanics , relativity , and quantum mechanics

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6188-521: Is a sequence of terms of decreasing nonnegative real numbers that converges to zero, and ( λ n ) {\displaystyle (\lambda _{n})} is a sequence of terms with bounded partial sums, then the series ∑ λ n a n {\textstyle \sum \lambda _{n}a_{n}} converges. Taking λ n = ( − 1 ) n {\displaystyle \lambda _{n}=(-1)^{n}} recovers

6307-501: Is also convergent, a property called absolute convergence . Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem . A historically important example of conditional convergence

6426-673: Is an effective way to prove convergence or absolute convergence of a series. For example, the series 1 + 1 4 + 1 9 + ⋯ + 1 n 2 + ⋯ {\textstyle 1+{\frac {1}{4}}+{\frac {1}{9}}+\cdots +{\frac {1}{n^{2}}}+\cdots \,} is convergent and absolutely convergent because 1 n 2 ≤ 1 n − 1 − 1 n {\textstyle {\frac {1}{n^{2}}}\leq {\frac {1}{n-1}}-{\frac {1}{n}}} for all n ≥ 2 {\displaystyle n\geq 2} and

6545-402: Is an ordered list. Like a set , it contains members (also called elements , or terms ). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers . One of the most important properties of

6664-418: Is associative, commutative, and distributes over series addition. Together with series addition, series multiplication gives the sets of absolutely convergent series of real numbers or complex numbers the structure of a commutative ring , and together with scalar multiplication as well, the structure of a commutative algebra ; these operations also give the sets of all series of real numbers or complex numbers

6783-558: Is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law , the Schrödinger equation , and the Einstein field equations . Functional analysis is also a major factor in quantum mechanics . When processing signals, such as audio , radio waves , light waves, seismic waves , and even images, Fourier analysis can isolate individual components of

6902-447: Is convergent per the alternating series test (and its sum is equal to  ln ⁡ 2 {\displaystyle \ln 2} ), though the series formed by taking the absolute value of each term is the ordinary harmonic series , which is divergent. The alternating series test can be viewed as a special case of the more general Dirichlet's test : if ( a n ) {\displaystyle (a_{n})}

7021-547: Is finite. Using comparisons to flattened-out versions of a series leads to Cauchy's condensation test : if the sequence of terms a n {\displaystyle a_{n}} is non-negative and non-increasing, then the two series ∑ a n {\textstyle \sum a_{n}} and ∑ 2 k a ( 2 k ) {\textstyle \sum 2^{k}a_{(2^{k})}} are either both convergent or both divergent. A series of real or complex numbers

7140-491: Is given by the termwise product c a 0 + c a 1 + c a 2 + ⋯ {\textstyle ca_{0}+ca_{1}+ca_{2}+\cdots } , or, in summation notation, c ∑ k = 0 ∞ a k = ∑ k = 0 ∞ c a k . {\displaystyle c\sum _{k=0}^{\infty }a_{k}=\sum _{k=0}^{\infty }ca_{k}.} Using

7259-538: Is given by the termwise sum ( a 0 + b 0 ) + ( a 1 + b 1 ) + ( a 2 + b 2 ) + ⋯ {\textstyle (a_{0}+b_{0})+(a_{1}+b_{1})+(a_{2}+b_{2})+\cdots \,} , or, in summation notation, ∑ k = 0 ∞ a k + ∑ k = 0 ∞ b k = ∑ k = 0 ∞

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7378-515: Is known or postulated. This is illustrated in classical mechanics , where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion ) may be solved explicitly. A measure on

7497-409: Is not as simple to establish as for addition. However, if both series a 0 + a 1 + a 2 + ⋯ {\displaystyle a_{0}+a_{1}+a_{2}+\cdots } and b 0 + b 1 + b 2 + ⋯ {\displaystyle b_{0}+b_{1}+b_{2}+\cdots } are absolutely convergent series, then

7616-499: Is now called naive set theory , and Baire proved the Baire category theorem . In the early 20th century, calculus was formalized using an axiomatic set theory . Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration , which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations . The idea of normed vector space

7735-459: Is now known as Rolle's theorem . In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series , of functions such as sine , cosine , tangent and arctangent . Alongside his development of Taylor series of trigonometric functions , he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at

7854-404: Is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions ). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation , complex analysis is widely applicable to two-dimensional problems in physics . Functional analysis is a branch of mathematical analysis, the core of which is formed by

7973-467: Is said to be conditionally convergent (or semi-convergent ) if it is convergent but not absolutely convergent. Conditional convergence is tested for differently than absolute convergence. One important example of a test for conditional convergence is the alternating series test or Leibniz test : A series of the form ∑ ( − 1 ) n a n {\textstyle \sum (-1)^{n}a_{n}} with all

8092-467: Is the alternating harmonic series , ∑ n = 1 ∞ ( − 1 ) n + 1 n = 1 − 1 2 + 1 3 − 1 4 + 1 5 − ⋯ , {\displaystyle \sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}=1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots ,} which has

8211-542: Is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations ) for the problems of mathematical analysis (as distinguished from discrete mathematics ). Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Numerical analysis naturally finds applications in all fields of engineering and

8330-481: The a n {\displaystyle a_{n}} vary in sign. Using comparisons to geometric series specifically, those two general comparison tests imply two further common and generally useful tests for convergence of series with non-negative terms or for absolute convergence of series with general terms. First is the ratio test : if there exists a constant C < 1 {\displaystyle C<1} such that |

8449-458: The n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . For instance, the Lebesgue measure of the interval [ 0 , 1 ] {\displaystyle \left[0,1\right]} in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically,

8568-522: The Ancient Greeks , the idea that a potentially infinite summation could produce a finite result was considered paradoxical , most famously in Zeno's paradoxes . Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes , for instance in the quadrature of the parabola . The mathematical side of Zeno's paradoxes was resolved using the concept of

8687-465: The Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. This began when Fermat and Descartes developed analytic geometry , which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and

8806-408: The calculus of variations , ordinary and partial differential equations , Fourier analysis , and generating functions . During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of a mathematical function . Real analysis began to emerge as an independent subject when Bernard Bolzano introduced

8925-427: The prefix sum in computer science . The inverse transformation for recovering a sequence from its partial sums is the finite difference , another linear sequence transformation. Partial sums of series sometimes have simpler closed form expressions, for instance an arithmetic series has partial sums s n = ∑ k = 0 n ( a + k d ) =

9044-452: The real numbers or the field C {\displaystyle \mathbb {C} } of the complex numbers . If so, the set of all series is also itself a ring, one in which the addition consists of adding series terms together term by term and the multiplication is the Cauchy product . A series or, redundantly, an infinite series , is an infinite sum. It is often represented as

9163-441: The "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers . It is useful in many branches of mathematics, including algebraic geometry , number theory , applied mathematics ; as well as in physics , including hydrodynamics , thermodynamics , mechanical engineering , electrical engineering , and particularly, quantum field theory . Complex analysis

9282-480: The "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences , including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity , smoothness and related properties of real-valued functions. Complex analysis (traditionally known as

9401-581: The 17th century during the Scientific Revolution , but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics . For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy . (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of

9520-446: The attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. Also, various pathological objects , (such as nowhere continuous functions , continuous but nowhere differentiable functions , and space-filling curves ), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure , Cantor developed what

9639-520: The concept of the Cauchy sequence , and started the formal theory of complex analysis . Poisson , Liouville , Fourier and others studied partial differential equations and harmonic analysis . The contributions of these mathematicians and others, such as Weierstrass , developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration , and made significant advances in complex analysis. Towards

9758-586: The concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems , a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. From Jain literature, it appears that Hindus were in possession of

9877-439: The context of real and complex numbers and functions . Analysis evolved from calculus , which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry ; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space ) or specific distances between objects (a metric space ). Mathematical analysis formally developed in

9996-400: The difference between the sum of a series and its n {\displaystyle n} th partial sum, s − s n = ∑ k = n + 1 ∞ a k , {\textstyle s-s_{n}=\sum _{k=n+1}^{\infty }a_{k},} is known as the n {\displaystyle n} th truncation error of

10115-1989: The divergence of the harmonic series, so the alternating harmonic series is conditionally convergent. For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields 1 − 1 2 − 1 4 + 1 3 − 1 6 − 1 8 + 1 5 − 1 10 − 1 12 + ⋯ = ( 1 − 1 2 ) − 1 4 + ( 1 3 − 1 6 ) − 1 8 + ( 1 5 − 1 10 ) − 1 12 + ⋯ = 1 2 − 1 4 + 1 6 − 1 8 + 1 10 − 1 12 + ⋯ = 1 2 ( 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + ⋯ ) , {\displaystyle {\begin{aligned}&1-{\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{3}}-{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{5}}-{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[3mu]&\quad =\left(1-{\frac {1}{2}}\right)-{\frac {1}{4}}+\left({\frac {1}{3}}-{\frac {1}{6}}\right)-{\frac {1}{8}}+\left({\frac {1}{5}}-{\frac {1}{10}}\right)-{\frac {1}{12}}+\cdots \\[3mu]&\quad ={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{6}}-{\frac {1}{8}}+{\frac {1}{10}}-{\frac {1}{12}}+\cdots \\[3mu]&\quad ={\frac {1}{2}}\left(1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots \right),\end{aligned}}} which

10234-410: The empty set, countable unions , countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice . Numerical analysis

10353-489: The end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts , in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions . Around that time,

10472-431: The first creates the series 1 + ( − 1 + 1 ) + {\displaystyle 1+(-1+1)+{}} ( − 1 + 1 ) + ⋯ = {\displaystyle (-1+1)+\cdots ={}} 1 + 0 + 0 + ⋯ , {\displaystyle 1+0+0+\cdots ,} which has partial sums equal to one for every term and thus sums to one,

10591-483: The formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433  BCE . Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what

10710-456: The grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied in Oresme's proof of the divergence of

10829-392: The harmonic series , and it is the basis for the general Cauchy condensation test . In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of the commutativity of addition. a 0 + a 1 + a 2 = {\displaystyle a_{0}+a_{1}+a_{2}={}}

10948-474: The infinite series. An example of a convergent series is the geometric series 1 + 1 2 + 1 4 + 1 8 + ⋯ + 1 2 k + ⋯ . {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{k}}}+\cdots .} It can be shown by algebraic computation that each partial sum s n {\displaystyle s_{n}}

11067-645: The limit of the sequence of partial sums of the resulting series, satisfies lim n → ∞ s a + b , n = lim n → ∞ ( s a , n + s b , n ) = lim n → ∞ s a , n + lim n → ∞ s b , n , {\displaystyle \lim _{n\rightarrow \infty }s_{a+b,n}=\lim _{n\rightarrow \infty }(s_{a,n}+s_{b,n})=\lim _{n\rightarrow \infty }s_{a,n}+\lim _{n\rightarrow \infty }s_{b,n},} when

11186-419: The limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series is divergent, then any nonzero scalar multiple of it is also divergent. Scalar multiplication of real numbers and complex numbers is associative, commutative, invertible, and it distributes over series addition. In summary, series addition and scalar multiplication gives

11305-432: The limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series. The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times − 1 {\displaystyle -1} will yield

11424-520: The modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals . Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y . He also introduced

11543-605: The original. Newton–Euler equations express the dynamics of a rigid body. Euler has been credited with contributing to the rise of Newtonian mechanics especially in topics other than gravity. This article about a mathematical publication is a stub . You can help Misplaced Pages by expanding it . Analysis (math) Analysis is the branch of mathematics dealing with continuous functions , limits , and related theories, such as differentiation , integration , measure , infinite sequences , series , and analytic functions . These theories are usually studied in

11662-456: The other indefinitely—and, if the series is convergent, the sum of the series—the explicit limit of the process. This is a generalization of the similar convention of denoting by a + b {\displaystyle a+b} both the addition —the process of adding—and its result—the sum of a and b . Commonly, the terms of a series come from a ring , often the field R {\displaystyle \mathbb {R} } of

11781-427: The other. The study of series is a major part of calculus and its generalization, mathematical analysis . Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions . The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics , computer science , statistics and finance . Among

11900-411: The partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called conditionally convergent series. Those that converge to the same value regardless of rearrangement are called unconditionally convergent series. For series of real numbers and complex numbers,

12019-469: The physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Vector analysis , also called vector calculus ,

12138-412: The product of the n {\displaystyle n} first positive integers , and 0 ! {\displaystyle 0!} is conventionally equal to 1. {\displaystyle 1.} Given a series s = ∑ k = 0 ∞ a k {\textstyle s=\sum _{k=0}^{\infty }a_{k}} , its n th partial sum

12257-436: The sequence of its partial sums has a limit . When the limit of the sequence of partial sums does not exist, the series diverges or is divergent . When the limit of the partial sums exists, it is called the sum of the series or value of the series : ∑ k = 0 ∞ a k = lim n → ∞ ∑ k = 0 n

12376-438: The series ∑ n = 0 ∞ 1 n ! = 1 + 1 + 1 2 + 1 6 + ⋯ + 1 n ! + ⋯ , {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}=1+1+{\frac {1}{2}}+{\frac {1}{6}}+\cdots +{\frac {1}{n!}}+\cdots ,} where n ! {\displaystyle n!} denotes

12495-622: The series is convergent and converges to 2 with truncation errors 1 / 2 n {\textstyle 1/2^{n}} . By contrast, the geometric series ∑ k = 0 ∞ 2 k {\displaystyle \sum _{k=0}^{\infty }2^{k}} is divergent in the real numbers . However, it is convergent in the extended real number line , with + ∞ {\displaystyle +\infty } as its limit and + ∞ {\displaystyle +\infty } as its truncation error at every step. When

12614-704: The series resulting from multiplying them also converges absolutely with a sum equal to the product of the two sums of the multiplied series, lim n → ∞ s c , n = ( lim n → ∞ s a , n ) ⋅ ( lim n → ∞ s b , n ) . {\displaystyle \lim _{n\rightarrow \infty }s_{c,n}=\left(\,\lim _{n\rightarrow \infty }s_{a,n}\right)\cdot \left(\,\lim _{n\rightarrow \infty }s_{b,n}\right).} Series multiplication of absolutely convergent series of real numbers and complex numbers

12733-404: The series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum of a 0 +

12852-409: The series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements, a n = s n − s n − 1 . {\displaystyle a_{n}=s_{n}-s_{n-1}.} Partial summation of a sequence is an example of a linear sequence transformation , and it is also known as

12971-598: The set of convergent series and the set of series of real numbers the structure of a real vector space . Similarly, one gets complex vector spaces for series and convergent series of complex numbers. All these vector spaces are infinite dimensional. The multiplication of two series a 0 + a 1 + a 2 + ⋯ {\displaystyle a_{0}+a_{1}+a_{2}+\cdots } and b 0 + b 1 + b 2 + ⋯ {\displaystyle b_{0}+b_{1}+b_{2}+\cdots } to generate

13090-406: The sets of all series of real numbers or complex numbers (regardless of convergence properties) the structure of an abelian group. The product of a series a 0 + a 1 + a 2 + ⋯ {\textstyle a_{0}+a_{1}+a_{2}+\cdots } with a constant number c {\displaystyle c} , called a scalar in this context,

13209-451: The simplest tests for convergence of a series, applicable to all series, is the vanishing condition or n th-term test : If lim n → ∞ a n ≠ 0 {\textstyle \lim _{n\to \infty }a_{n}\neq 0} , then the series diverges; if lim n → ∞ a n = 0 {\textstyle \lim _{n\to \infty }a_{n}=0} , then

13328-2463: The structure of an associative algebra . ∑ n = 1 ∞ 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ = π 2 6 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}} ∑ n = 1 ∞ ( − 1 ) n + 1 ( 4 ) 2 n − 1 = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − 4 11 + 4 13 − ⋯ = π {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}(4)}{2n-1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots =\pi } ∑ n = 1 ∞ ( − 1 ) n + 1 n = ln ⁡ 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2} ∑ n = 1 ∞ 1 2 n n = ln ⁡ 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}n}}=\ln 2} ∑ n = 0 ∞ ( − 1 ) n n ! = 1 − 1 1 ! + 1 2 ! − 1 3 ! + ⋯ = 1 e {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}=1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots ={\frac {1}{e}}} ∑ n = 0 ∞ 1 n ! = 1 0 ! + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + ⋯ = e {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =e} One of

13447-570: The study of differential and integral equations . Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves . This includes the study of the notions of Fourier series and Fourier transforms ( Fourier analysis ), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory , number theory , representation theory , signal processing , quantum mechanics , tidal analysis , and neuroscience . A differential equation

13566-625: The study of vector spaces endowed with some kind of limit-related structure (e.g. inner product , norm , topology , etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous , unitary etc. operators between function spaces. This point of view turned out to be particularly useful for

13685-407: The sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement. However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of

13804-408: The symbols s a , n {\displaystyle s_{a,n}} for the partial sums of the original series and s c a , n {\displaystyle s_{ca,n}} for the partial sums of the series after multiplication by c {\displaystyle c} , this definition implies that s c a , n = c s

13923-480: The tangents of curves. Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system , is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus , which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as

14042-429: The test is inconclusive. When every term of a series is a non-negative real number, for instance when the terms are the absolute values of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound for a series or for the absolute values of its terms

14161-412: Was in the air, and in the 1920s Banach created functional analysis . In mathematics , a metric space is a set where a notion of distance (called a metric ) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line , the complex plane , Euclidean space , other vector spaces , and the integers . Examples of analysis without

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