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54-544: (Redirected from P-series ) P series or P-series may refer to: the p -series in mathematics, related to convergence of certain series P-series fuels , blends of fuels Huawei P series , mobile phone series by Huawei IBM pSeries , computer series by IBM Ruger P series – pistols ThinkPad P series , mobile workstation line by Lenovo Sony Cybershot P-series digital cameras, see Cyber-shot Sony Vaio P series – notebook computers Sony Ericsson P series ,

108-507: A n {\displaystyle A=\sum _{n=1}^{\infty }a_{n}} converges if and only if the sum A ∗ = ∑ n = 0 ∞ 2 n a 2 n {\displaystyle A^{*}=\sum _{n=0}^{\infty }2^{n}a_{2^{n}}} converges. Moreover, if they converge, then A ≤ A ∗ ≤ 2 A {\displaystyle A\leq A^{*}\leq 2A} holds. Suppose

162-462: A n {\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}a_{n}} are convergent series. This test is also known as the Leibniz criterion . If { a n } {\displaystyle \{a_{n}\}} is a sequence of real numbers and { b n } {\displaystyle \{b_{n}\}} a sequence of complex numbers satisfying where M

216-401: A n {\displaystyle \sum _{n=1}^{\infty }a_{n}} be an infinite series with real terms and let f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } be any real function such that f ( 1 / n ) = a n {\displaystyle f(1/n)=a_{n}} for all positive integers n and

270-582: A n {\displaystyle \sum a_{n}} diverges if and only if there is a sequence b n {\displaystyle b_{n}} of positive numbers such that b k ( a k / a k + 1 ) − b k + 1 ≤ 0 {\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\leq 0} and ∑ 1 / b n {\displaystyle \sum 1/b_{n}} diverges. Let ∑ n = 1 ∞

324-432: A n ) n ≥ 1 {\displaystyle (a_{n})_{n\geq 1}} and ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} be two sequences of real numbers. Assume that ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} is a strictly monotone and divergent sequence and

378-417: A n b n {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}} exists, is finite and non-zero, then either both series converge or both series diverge. Let { a n } {\displaystyle \left\{a_{n}\right\}} be a non-negative non-increasing sequence. Then the sum A = ∑ n = 1 ∞

432-488: A n < 1 {\displaystyle 0<a_{n}<1} holds, then ∏ n = 1 ∞ ( 1 − a n ) {\displaystyle \prod _{n=1}^{\infty }(1-a_{n})} approaches a non-zero limit if and only if the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges . This can be proved by taking

486-457: A n } be a sequence of positive numbers. Define If exists there are three possibilities: An alternative formulation of this test is as follows. Let { a n } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that for all n > K then the series { a n } is convergent. Let { a n } be a sequence of positive numbers. Define If exists, there are three possibilities: Let {

540-471: A n } be a sequence of positive numbers. If a n a n + 1 = 1 + α n + O ( 1 / n β ) {\displaystyle {\frac {a_{n}}{a_{n+1}}}=1+{\frac {\alpha }{n}}+O(1/n^{\beta })} for some β > 1, then ∑ a n {\displaystyle \sum a_{n}} converges if α > 1 and diverges if α ≤ 1 . Let {

594-554: A n } be a sequence of positive numbers. Then: (1) ∑ a n {\displaystyle \sum a_{n}} converges if and only if there is a sequence b n {\displaystyle b_{n}} of positive numbers and a real number c > 0 such that b k ( a k / a k + 1 ) − b k + 1 ≥ c {\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\geq c} . (2) ∑

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648-765: A subset of the real numbers with real values is called monotonic if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed monotonically increasing (also increasing or non-decreasing ) if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\!\left(x\right)\leq f\!\left(y\right)} , so f {\displaystyle f} preserves

702-679: A (possibly non-linear) operator T : X → X ∗ {\displaystyle T:X\rightarrow X^{*}} is said to be a monotone operator if ( T u − T v , u − v ) ≥ 0 ∀ u , v ∈ X . {\displaystyle (Tu-Tv,u-v)\geq 0\quad \forall u,v\in X.} Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives. A subset G {\displaystyle G} of X × X ∗ {\displaystyle X\times X^{*}}

756-488: A generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total . Furthermore, the strict relations < {\displaystyle <} and > {\displaystyle >} are of little use in many non-total orders and hence no additional terminology

810-486: A monotonic transform (see also monotone preferences ). In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers. The following properties are true for a monotonic function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } : These properties are

864-506: A non-negative and monotonically decreasing function such that f ( n ) = a n {\displaystyle f(n)=a_{n}} . If ∫ 1 ∞ f ( x ) d x = lim t → ∞ ∫ 1 t f ( x ) d x < ∞ , {\displaystyle \int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,} then

918-451: A one-to-one mapping from their range to their domain. However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one). A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if y = g ( x ) {\displaystyle y=g(x)}

972-445: A sequence of positive numbers. Then the infinite product ∏ n = 1 ∞ ( 1 + a n ) {\displaystyle \prod _{n=1}^{\infty }(1+a_{n})} converges if and only if the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges. Also similarly, if 0 <

1026-413: A series of cell phones Vespa P-series motor scooters See also [ edit ] O series (disambiguation) Q series (disambiguation) T series (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title P series . If an internal link led you here, you may wish to change the link to point directly to

1080-422: A source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible. The term monotonic transformation (or monotone transformation ) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across

1134-412: Is lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test , test for divergence , or

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1188-468: Is a monotonically increasing function. A function is unimodal if it is monotonically increasing up to some point (the mode ) and then monotonically decreasing. When f {\displaystyle f} is a strictly monotonic function, then f {\displaystyle f} is injective on its domain, and if T {\displaystyle T} is the range of f {\displaystyle f} , then there

1242-479: Is also admissible , monotonicity is a stricter requirement than admissibility. Some heuristic algorithms such as A* can be proven optimal provided that the heuristic they use is monotonic. In Boolean algebra , a monotonic function is one such that for all a i and b i in {0,1} , if a 1 ≤ b 1 , a 2 ≤ b 2 , ..., a n ≤ b n (i.e. the Cartesian product {0, 1}

1296-439: Is also monotone. The dual notion is often called antitone , anti-monotone , or order-reversing . Hence, an antitone function f satisfies the property x ≤ y ⟹ f ( y ) ≤ f ( x ) , {\displaystyle x\leq y\implies f(y)\leq f(x),} for all x and y in its domain. A constant function is both monotone and antitone; conversely, if f

1350-472: Is an inverse function on T {\displaystyle T} for f {\displaystyle f} . In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse. The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the y -axis. A map f : X → Y {\displaystyle f:X\to Y}

1404-604: Is both monotone and antitone, and if the domain of f is a lattice , then f must be constant. Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y {\displaystyle x\leq y} if and only if f ( x ) ≤ f ( y ) ) {\displaystyle f(x)\leq f(y))} and order isomorphisms ( surjective order embeddings). In

1458-826: Is called strictly monotone . Functions that are strictly monotone are one-to-one (because for x {\displaystyle x} not equal to y {\displaystyle y} , either x < y {\displaystyle x<y} or x > y {\displaystyle x>y} and so, by monotonicity, either f ( x ) < f ( y ) {\displaystyle f\!\left(x\right)<f\!\left(y\right)} or f ( x ) > f ( y ) {\displaystyle f\!\left(x\right)>f\!\left(y\right)} , thus f ( x ) ≠ f ( y ) {\displaystyle f\!\left(x\right)\neq f\!\left(y\right)} .) To avoid ambiguity,

1512-476: Is finitely convergent if and only if α > 1 {\displaystyle \alpha >1} . While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products . This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be

1566-491: Is introduced for them. Letting ≤ {\displaystyle \leq } denote the partial order relation of any partially ordered set, a monotone function, also called isotone , or order-preserving , satisfies the property x ≤ y ⟹ f ( x ) ≤ f ( y ) {\displaystyle x\leq y\implies f(x)\leq f(y)} for all x and y in its domain. The composite of two monotone mappings

1620-487: Is neither non-decreasing nor non-increasing. A function f {\displaystyle f} is said to be absolutely monotonic over an interval ( a , b ) {\displaystyle \left(a,b\right)} if the derivatives of all orders of f {\displaystyle f} are nonnegative or all nonpositive at all points on the interval. All strictly monotonic functions are invertible because they are guaranteed to have

1674-524: Is ordered coordinatewise ), then f( a 1 , ..., a n ) ≤ f( b 1 , ..., b n ) . In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an n -ary Boolean function is monotonic when its representation as an n -cube labelled with truth values has no upward edge from true to false . (This labelled Hasse diagram

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1728-403: Is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator G ( T ) {\displaystyle G(T)} is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set . Order theory deals with arbitrary partially ordered sets and preordered sets as

1782-472: Is said to be monotone if each of its fibers is connected ; that is, for each element y ∈ Y , {\displaystyle y\in Y,} the (possibly empty) set f − 1 ( y ) {\displaystyle f^{-1}(y)} is a connected subspace of X . {\displaystyle X.} In functional analysis on a topological vector space X {\displaystyle X} ,

1836-553: Is said to be a monotone set if for every pair [ u 1 , w 1 ] {\displaystyle [u_{1},w_{1}]} and [ u 2 , w 2 ] {\displaystyle [u_{2},w_{2}]} in G {\displaystyle G} , ( w 1 − w 2 , u 1 − u 2 ) ≥ 0. {\displaystyle (w_{1}-w_{2},u_{1}-u_{2})\geq 0.} G {\displaystyle G}

1890-426: Is some constant, then the series converges. A series ∑ i = 0 ∞ a i {\displaystyle \sum _{i=0}^{\infty }a_{i}} is convergent if and only if for every ε > 0 {\displaystyle \varepsilon >0} there is a natural number N such that holds for all n > N and all p ≥ 1 . Let (

1944-403: Is strictly increasing on the range [ a , b ] {\displaystyle [a,b]} , then it has an inverse x = h ( y ) {\displaystyle x=h(y)} on the range [ g ( a ) , g ( b ) ] {\displaystyle [g(a),g(b)]} . The term monotonic is sometimes used in place of strictly monotonic , so

1998-417: Is the dual of the function's labelled Venn diagram , which is the more common representation for n ≤ 3 .) The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular not is forbidden). For instance "at least two of a , b , c hold" is a monotonic function of

2052-602: The Riemann zeta function applied to p {\displaystyle p} , that is ζ ( p ) {\displaystyle \zeta (p)} . If the series ∑ n = 1 ∞ b n {\displaystyle \sum _{n=1}^{\infty }b_{n}} is an absolutely convergent series and | a n | ≤ | b n | {\displaystyle |a_{n}|\leq |b_{n}|} for sufficiently large n  , then

2106-402: The context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions . A heuristic h ( n ) {\displaystyle h(n)} is monotonic if, for every node n and every successor n' of n generated by any action a , the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus

2160-566: The divergence test . This is also known as d'Alembert's criterion . This is also known as the n th root test or Cauchy's criterion . The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. The series can be compared to an integral to establish convergence or divergence. Let f : [ 1 , ∞ ) → R + {\displaystyle f:[1,\infty )\to \mathbb {R} _{+}} be

2214-418: The estimated cost of reaching the goal from n' , h ( n ) ≤ c ( n , a , n ′ ) + h ( n ′ ) . {\displaystyle h(n)\leq c\left(n,a,n'\right)+h\left(n'\right).} This is a form of triangle inequality , with n , n' , and the goal G n closest to n . Because every monotonic heuristic

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2268-467: The following limit exists: Then, the limit Suppose that ( f n ) is a sequence of real- or complex-valued functions defined on a set A , and that there is a sequence of non-negative numbers ( M n ) satisfying the conditions Then the series converges absolutely and uniformly on A . The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case. Let {

2322-565: The following statements are true: Then ∑ a n b n {\displaystyle \sum a_{n}b_{n}} is also convergent. Every absolutely convergent series converges. Suppose the following statements are true: Then ∑ n = 1 ∞ ( − 1 ) n a n {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}} and ∑ n = 1 ∞ ( − 1 ) n + 1

2376-461: The harmonic series, which diverges. The case of p = 2 , k = 1 {\displaystyle p=2,k=1} is the Basel problem and the series converges to π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}} . In general, for p > 1 , k = 1 {\displaystyle p>1,k=1} , the series is equal to

2430-409: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=P_series&oldid=1101772348 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Convergence tests#p-series test If the limit of the summand is undefined or nonzero, that

2484-445: The logarithm of the product and using limit comparison test. Monotonic function In mathematics , a monotonic function (or monotone function ) is a function between ordered sets that preserves or reverses the given order . This concept first arose in calculus , and was later generalized to the more abstract setting of order theory . In calculus , a function f {\displaystyle f} defined on

2538-463: The order ≤ {\displaystyle \leq } in the definition of monotonicity is replaced by the strict order < {\displaystyle <} , one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing ). Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing ). A function with either property

2592-406: The order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing ) if, whenever x ≤ y {\displaystyle x\leq y} , then f ( x ) ≥ f ( y ) {\displaystyle f\!\left(x\right)\geq f\!\left(y\right)} , so it reverses the order (see Figure 2). If

2646-511: The reason why monotonic functions are useful in technical work in analysis . Other important properties of these functions include: An important application of monotonic functions is in probability theory . If X {\displaystyle X} is a random variable , its cumulative distribution function F X ( x ) = Prob ( X ≤ x ) {\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}

2700-479: The second derivative f ″ {\displaystyle f''} exists at x = 0 {\displaystyle x=0} . Then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges absolutely if f ( 0 ) = f ′ ( 0 ) = 0 {\displaystyle f(0)=f'(0)=0} and diverges otherwise. Consider

2754-433: The series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges absolutely. If { a n } , { b n } > 0 {\displaystyle \{a_{n}\},\{b_{n}\}>0} , (that is, each element of the two sequences is positive) and the limit lim n → ∞

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2808-437: The series Cauchy condensation test implies that ( i ) is finitely convergent if is finitely convergent. Since ( ii ) is a geometric series with ratio 2 ( 1 − α ) {\displaystyle 2^{(1-\alpha )}} . ( ii ) is finitely convergent if its ratio is less than one (namely α > 1 {\displaystyle \alpha >1} ). Thus, ( i )

2862-765: The series converges. But if the integral diverges, then the series does so as well. In other words, the series a n {\displaystyle {a_{n}}} converges if and only if the integral converges. A commonly-used corollary of the integral test is the p-series test. Let k > 0 {\displaystyle k>0} . Then ∑ n = k ∞ ( 1 n p ) {\displaystyle \sum _{n=k}^{\infty }{\bigg (}{\frac {1}{n^{p}}}{\bigg )}} converges if p > 1 {\displaystyle p>1} . The case of p = 1 , k = 1 {\displaystyle p=1,k=1} yields

2916-442: The terms weakly monotone , weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity. The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it

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