What is Alternating series: Definition and 110 Discussions

In mathematics, an alternating series is an infinite series of the form








{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}








{\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}}
with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.

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  1. C

    Ratio Test vs AST

    Hi, I'm having difficulty understanding why the interval of convergence is (0, 18]. When I tested x=18, I got the following conclusion using the ratio test. When I attempt using AST, the function still diverges as the lim (n -> inf) = 2^n / n ≠ 0. What am I missing? Thanks!
  2. murshid_islam

    Does the Alternating Series Test show convergence for this series?

    The Alternating series test has to be used to determine whether this series converges or diverges: \sum\limits_{n=1}^{\infty} (-1)^n\frac{\sqrt n}{2n+3} Here's what I have done: Let a_n = \frac{\sqrt n}{2n+3}. Therefore, a_{n+1} = \frac{\sqrt {n+1}}{2n+5} Now, for a_{n+1} to be less than or...
  3. karush

    MHB 10.6.2 converge or diverge? alternating series

    converge or diverge $$S_n= \sum_{n=1}^{\infty} (-1)^{n+1}\frac{\sqrt{n}+6}{n+4}$$ ok by graph the first 10 terms it looks alterations are converging to 0
  4. Matt Benesi

    B Generating the formula for the coefficients of an alternating series

    ETA. Read the bottom post first. Well, and.. Obviously mathematicians know this identity. At the x=b=c=n=2 point, pi exists. There are also connections to the Wallis product (pi/2). Anyway, I simplified it to the n=2 case. And re-remembered my fascination with the Pidentity, where...
  5. Matt Benesi

    B Periodic smooth alternating series other than sin and cos

    1) Are there any periodic alternating series functions other than sine and cosine (and series derived from them, like the series for cos(a) * cos(b))? 2) What is the following series called when x is (0,1) and (1,2]? Quasiperiodic? Semi? \sum_{n=0}^\infty \, (-1)^n \...
  6. Mr Davis 97

    I Proof of Alternating Series Test

    I'm looking at the proof of the alternating series test, and the basic idea is that the odd and even partial sums converge to the same number, and that this implies that the series converges as a whole. What I don't understand is why the even and odd partial sums converging to the same limit...
  7. jlmccart03

    Alternating Series Estimation Theorem

    Homework Statement Using the power series for ln(x + 1) and the Estimation Theorem for the Alternating Series, we conclude that the least number of terms in the series needed to approximate ln 2 with error < 3/1000 is: (i) 333 (ii) 534 (iii) 100 (iv) 9 (v) 201 Homework Equations ln(x+1) =...
  8. PhysicsCollegeGirl

    Master Power Series Convergence with Expert Help - Examples Included

    Homework Statement [/B] There are three problems that I am struggling with. 1. ∑[k2(x-2)k]/[3k] 2. ∑[(x-4)n]/[(n)(-9)n] 3. ∑[2k(x-3)k]/[k(k+1)] The Attempt at a Solution On the first two I am having problems finding the end-points of the interval of convergence. I use the ratio test. 1...
  9. solour

    Why does (-1)^n(sin(pi/n)) converge when (sin(p/n)) diverges

    Homework Statement I know that ∑n=1 to infinity (sin(p/n)) diverges due using comparison test with pi/n, despite it approaching 0 as n approaches infinity. However, an alternating series with (-1)^n*sin(pi/n) converges. Which does not make sense because it consists of two diverging functions...
  10. B

    Absolutely Convergent, Conditionally Convergent, or Divergent?

    Homework Statement ∞ Σ (-1)n-1 n/n2 +4 n=1 Homework Equations lim |an+1/an| = L n→∞ bn+1≤bn lim bn = 0 n→∞ The Attempt at a Solution So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo. I found that...
  11. H

    I Convergence of an alternating series

    Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##. If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why? This is not explained in the following proof:
  12. Drakkith

    I Alternating Series, Testing for Convergence

    The criteria for testing for convergence with the alternating series test, according to my book, is: Σ(-1)n-1bn With bn>0, bn+1 ≤ bn for all n, and lim n→∞bn = 0. My question is about the criteria. I'm running into several homework problem where bn is not always greater than bn+1, such as the...
  13. I

    Convergence of alternating series

    Homework Statement Do the following series converge or diverge? ## \sum_{n=2}^\infty \frac{1}{\sqrt{n} +(-1)^nn}## and ##\sum_{n=2}^\infty \frac{1}{1+(-1)^n\sqrt{n}}##. Homework Equations Leibniz convergence criteria: If ##\{a_n\}_{k=1}^\infty## is positive, decreasing and ##a_n \to 0##, the...
  14. M

    Alternating series test for convergence

    Homework Statement Homework Equations The Attempt at a Solution I don't get how they got what's stated in the above picture. Where does 1/2 and n/(n + 1) come from? Can't you just show that an + 1 ≤ an?
  15. Abscissas

    Alternative examples, alternating series test

    Hey guys, this one is just for funnsies. So when dealing with an alternating series test, 3 requirements must be met, : Alternating u(sub n) ≥ u(sub n+1) for all n ≥ N, for some integer N u(sub n) → 0 as n → ∞. So I have been coming up with examples where of these are true, and one isnt. A...
  16. J

    How to deal with (ln(x))^p in an Alternating Series Test

    Homework Statement Determine all values of P for which the series ∑((-1)^(n-1))((ln(x))^p)/(5n) is convergent, expressing your answer in interval notation (Problem is shown in attached picture). Homework Equations Alternating Series Test: If {a_n} is positive and decreasing, and if the lim as...
  17. M

    Can an alternating series with decreasing terms converge to zero?

    Hi PF! The other day I was showing convergence for an alternating series, let's call it ##\sum (-1)^n b_n##. I showed that ##\lim_{n \to \infty} b_n = 0## and that ##b_n## was monotonically decreasing; hence the series converges by the alternating series test. but I needed also to show it did...
  18. A

    Factorials within alternating series

    Homework Statement ∑ [ (-1)^n * n!/(10^n) ] 2. The attempt at a solution the problem is that I cannot use derivative to make sure that a(n) is decreasing neither L hopital rule to find the limit.
  19. RJLiberator

    Find the Sum of this Alternating Series

    Homework Statement Find the sum of starts at 0 to infinity ∑ (cos(k*pi))/pi^k First, I determined that it does, indeed, converge with the alternating series test. Second, I found the answer to be pi/(1+pi) via wolfram alpha. But I am at a loss on how to find the answer here. This is a...
  20. J

    Calc II - Alternating Series Test/Limits

    Hello PF, I've got a homework question I'm having some trouble with regarding series, particularily alternating series. The question asks you to test the series for convergence or divergence for an alternate series by using the A.S.T. : ∞ ∑ (-1)n-1e2/n n=1 Homework Equations...
  21. jdawg

    Alternating Series Test No Divergence?

    Homework Statement Hey! So I just have a quick question. In my notes I wrote down that the alternating series test only proves absolute or conditional convergence, but can not prove divergence. Is this true or did I misunderstand my professor? Homework Equations The Attempt at a...
  22. jdawg

    Alternating series test problem

    Homework Statement ∞n=1∑(-1)n\stackrel{10n}{(n+1)!} Homework Equations The Attempt at a Solution I already found that the limit does equal zero by using the ratio test on bn. What I'm having trouble with is determining if it decreases or not. I know you can't take the derivative...
  23. alyafey22

    MHB Proving $\log(2)$ with Alternating Series

    It might be well-known for you that \sum_{n\geq 1}\frac{(-1)^{n+1}}{n}=\log(2) There might be more than one way to prove it :)
  24. J

    Alternating Series: Solving Homework Equations

    Homework Statement Ʃ (-1)^n [ n+ln(n) / n-ln(n)] from n = 2 to infinity. Homework Equations I looked at the limit first because I thought lnn was very slow function. n would go faster. The Attempt at a Solution limit n --> ∞ [ n+ln(n) / n-ln(n)] = 1 so it diverges. Limit is...
  25. F

    Alternating Series estimation theorem vs taylor remainder

    Homework Statement Let Tn(x) be the degree n polynomial of the function sin x at a=0. Suppose you approx f(x) by Tn(x) if abs(x)<=1, how many terms are need (what is n) to obtain an error less than 1/120 Homework Equations Rn(x)=M(x-a)^(n+1)/(n+1)! sin(x)=sum from 0 to ∞ of...
  26. Lebombo

    Alternating Series Test Conditions

    Homework Statement This is what I understand about Alternating Series right now: If I have an alternate series, I can apply the alternative series test. \sum(-1)^{n}a_{n} Condition 1: Nth term test on a_{n} Condition 2: 0 < a_{n+1} ≤ a_{n}If condition 1 is positive or ∞, convergence is...
  27. Seydlitz

    Proof of the Alternating Series Approximation Theorem

    Homework Statement Problem taken from Boas Mathematical Methods book, Section 14 page 35. Prove that if ##S=\sum_{n=1}^{\infty} a_n## is an alternating series with ##|a_{n+1}|<|a_n|##, and ##\lim_{n \to \infty} a_n=0##, then ##|S-(a_1+a_2+...+a_n)|\leq|a_{n+1}|##. The Attempt at a...
  28. C

    Understanding this proof involving alternating series

    I'm having trouble the underlined red part of this proof (attached image) of the what looks to be the alternate series test, not sure if it's an error but it's more likely I've perhaps misunderstood something. If y_j is defined as the sequence of partial sums of the even terms of the sequence...
  29. J

    Alternating series something dissapeared.

    Homework Statement Hi, The question wanted to know if the alternating series converges or diverges. $$A_n = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n+1}$$ Homework Equations [b]3. The Attempt at a Solution [/b You can see it here ...
  30. F

    Alternating Series Test - No B_n?

    Homework Statement Ʃ(-1/2)^k from 0 to infinity. Homework Equations Ʃ(-1)^k*B_n from 0 to infinity where if the series converges 1. lim of B_n as n goes to infinity must = 0 2. B_n must be decreasing The Attempt at a Solution It doesn't look like there is a B_n in the original equation...
  31. C

    Alternating Series Convergence Test

    According to my calculus book two parts to testing an alternating series for convergence. Let s = Ʃ(-1)n bn. The first is that bn + 1 < bn. The second is that the limn\rightarrow∞ bn = 0. However, isn't the first condition unnecessary since bn must be decreasing if the limit is zero. I...
  32. T

    Does the Series Sum of 1 + (-1)^n/n Converge?

    Homework Statement The problem contained five answer choices, of which I the answerer was to find one that fit the criteria of the question. The question was: "Which series of the following terms would be convergent?". It listed five series, The answer was this term: 1 + (-1)n / n. Homework...
  33. W

    What is the next step for determining convergence or divergence of this series?

    Homework Statement determine either absolute convergence, conditional convergence, or divergence for the series.Homework Equations \displaystyle \sum^{∞}_{n=1} (-1)^n \frac{6n^8 + 3}{3n^5 + 3} The Attempt at a Solution I cannot use the alternating series test since the function is increasing...
  34. H

    When can you not apply the alternating series test?

    Homework Statement I have a series Ʃ(1 to infinity) ((-1)^n*n^n)/n! Homework Equations The Attempt at a Solution apparently you cannot use the alternating series for this question, why is this? It has the (-1)^n, what else is needed to allow you to use the alternating...
  35. G

    Nth term of alternating series with two positive terms at beginning

    (I hope this is the right subforum) I'm talking about the series 1, 1, -1, 1, -1, 1, -1... I thought about it for a long time but I have no idea. If that first term were gone it would just be (-1)^(n+1), but...it's there...
  36. D

    Sum of alternating series using four-digit chopping arithmetic

    Homework Statement Let a_{n} be an alternating series whose terms are decreasing in magnitude. How to compute the sum as precisely as possible using four-digit chopping arithmetic? In particular, apply the method to compute \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{(2n)!}}} and...
  37. A

    Alternating Series Test/Test for Divergence

    So I've been practicing several series that can be solved using the alternating series test, but I've came to a question that's been bothering me for sometime now. If a series fails the alternating series test, will the test for divergence always prove it to be divergent? Typically, in...
  38. M

    Proving Whether an Alternating Series is Divergent or Convergent

    Homework Statement Determine an explicit function for this sequence and determine whether it is convergent. an={1, 0, -1, 0, 1, 0, -1, 0, 1, ...} The Attempt at a Solution I came up with this function: an = cos(nπ/2), and wrote that as sigma notation from n=0 to infinity. Is...
  39. T

    Alternating Series Test for Convergence

    Homework Statement Does this series converge absolutely or conditionally?Homework Equations Series from n=1 to ∞ (-1)^(n+1) * n!/2^n The Attempt at a Solution In trying to apply the alternating series test, I have found the following: 1.) n!/2^n > 0 for n>0 2.) Next, in testing to see if...
  40. D

    Help understanding the conditions of the Alternating Series

    Homework Statement Okay from what I have learned to prove that a series converges via the alternating test, you must prove the following conditions Homework Equations 1) an > 0 2) lim an (n--> infinity) = 0 and 3) a(n+1) < an The Attempt at a Solution However recently I've been encountering...
  41. B

    Can the Alternating Series Test Prove Divergence?

    Prove that \sum^{∞}_{n=1}(-1)^{n} diverges. I realized that the alternating series test can only be used for convergence and not necessarily for divergence. I might have to apply a ε-δ proof (Yikes!) which I have never been good at so please help me out. BiP
  42. I'm Awesome

    Do Alternating Series Have Limits?

    I would imagine that an alternating series that goes of to infinity doesn't have a limit because it keeps switching back and forth, but I can't find anything in my textbook about it. I just want to make sure that this is right.
  43. DryRun

    Question about Alternating Series Test

    Homework Statement After reading a few topics on this forum, i just realized that i had misunderstood the sequence v/s series theorem when it concerns the alternating sign. So, i went back to my notes, and I'm surprised to see that there is no mention of a series test. There is only a sequence...
  44. C

    Calculus II - Alternating Series Test - Convergent?

    Hello! I was working some practice problems for a Calc II quiz for Friday on the alternating series test for convergence or divergence of a series. I ran into a problem when I was working the following series, trying to determine whether it was convergent or divergent: Homework Statement ∞...
  45. C

    A formula for approximating ln(2) and sums of other alternating series

    1 \ - \ \frac{1}{2} \ + \ \frac{1}{3} \ - \ \frac{1}{4} \ + \ ... \ - \ \frac{1}{n - 1} \ + \ \frac{1}{n} \ - \ \frac{1}{2n + 1} \ < \ ln(n), where n is a positive odd integer I worked this out (rediscovered it) and proved it by induction. For example, when n = 71...
  46. alexmahone

    MHB SE Class 12 Maths Alternating Series Test

    Alternating series test: If $\{a_n\}$ is positive and strictly decreasing, and $\lim a_n=0$, then $\sum(-1)^n a_n$ converges. Is the alternating series test still valid if "strictly decreasing" is omitted? Give a proof or counterexample.
  47. T

    Divergent alternating series problem

    Homework Statement If Ʃa_n is divergent, the absolute value of Ʃa_n is divergent. True or false. This is the main question I am trying to answer. I should be able to answer this problem on my own, but i ran into a problem that confused me. What I Did So I decided to start this...
  48. G

    How Can the Alternating Series Test Assume N=1?

    Alternating series test: 1. All the u_n are all positive 2. u_n\geq u_{n+1} for all n \geq N. For some integer N 3. u_n \rightarrow 0 I thought it would hold with 2. and that the su m of the N first terms were not \infty Here is the theroem just in case...
  49. D

    Alternating series (Leibniz criterion)

    I read that an alternating series \Sigma (-1)^n a_n converges if "and only if" the sequence a_n is both monotonous and converges to zero. I tried with this series: \Sigma_{n=1}^{\infty} (-1)^n | \frac{1}{n^2} \sin(n)| in the wolfram alpha and seems to converge to -0.61..., even if...
  50. T

    Alternating Series Tests: Understanding Conditional & Absolute Convergence

    I have a question about the ratio test. Suppose it proves inconclusive, we must than use another test to check for conditional convergence - 1) this test has to be associated with an alternating series, such as the Alternating Series Test, correct? (we wouldn't be able to use something like...