What is Series convergence: Definition and 111 Discussions

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series




{\displaystyle \sum _{n=1}^{\infty }a_{n}}

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  1. Drakkith

    I Infinite Series of Infinite Series

    I had a random thought about infinite series the other day while watching a math video. Let's say we have an infinite series where each term in the series is itself another infinite series. How would one go about finding the sum? For example, let's say we have the series ##a_1+a_2+a_3...##...
  2. P

    Series investigation: divergence/convergence

    Hi everyone! It's about the following task: show the convergence or divergence of the following series (combine estimates and criteria). I am not sure if I have solved the problem correctly. Can you guys help me? Is there anything I need to correct? I look forward to your feedback.
  3. W

    Series Convergence: What Can the Nth Term Test Tell Us?

    I'm not sure which test is the best to use, so I just start with a divergence test ##\lim_{n \to \infty} \frac {n+3}{\sqrt{5n^2+1}}## The +3 and +1 are negligible ##\lim_{n \to \infty} \frac {n}{\sqrt{5n^2}}## So now I have ##\infty / \infty##. So it's not conclusive. Trying ratio test...
  4. A

    Problem with series convergence — Taylor expansion of exponential

    Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!
  5. Coltrane8

    I Spherical Harmonics Expansion convergence

    In the contex of ##L^2## space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where ##Y_\ell^m( \theta , \varphi...
  6. mertcan

    How Can We Ensure Convergence in Function Approximations Beyond Taylor Series?

    Hi, as you know infinite sum of taylor series may not converge to its original function which means when we increase the degree of series then we may diverge more. Also you know taylor series is widely used for an approximation to vicinity of relevant point for any function. Let's think about a...
  7. T

    MHB Power Series Convergence Assistance

    The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.
  8. Tbonewillsone

    Recovering the delta function with sin⁡(nx)/x

    Homework Statement Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I...
  9. C

    MHB Series Convergence: Ratio Test & Lim. n→∞

    I'm trying to determine if \sum_{n=1}^{\infty}\frac{{n}^{10}}{{2}^{n}} converges or diverges. I did the ratio test but I'm left with determining \lim_{{n}\to{\infty}}\frac{(n+1)^{10}}{2n^{10}} Any suggestions??
  10. G

    I Trigonometric series with normalised coefficients

    Hi all, I have a trigonometric function series $$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$ with the normalization condition $$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$ and ##\Lambda_l## being monotonic decrescent...
  11. K

    Comparison test for series convergence (trig function)

    Homework Statement Use a comparison test to determine whether this series converges: \sum_{x=1}^{\infty }\sin ^2(\frac{1}{x}) Homework EquationsThe Attempt at a Solution At small values of x: \sin x\approx x a_{x}=\sin \frac{1}{x} b_{x}=\frac{1}{x} \lim...
  12. T

    MHB Series Convergence Or Divergence

    I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...
  13. evinda

    MHB Proving Series Convergence: Comparing $\sum y_n$ with $\sum \frac{y_n}{1+y_n}$

    Hello! (Wave) We have a sequence $(y_n)$ with $y_n \geq 0$. We assume that the series $\sum_{n=1}^{\infty} \frac{y_n}{1+y_n}$ converges. How can we show that the series $\sum_{n=1}^{\infty} y_n$ converges? It holds that $y_n \geq \frac{y_n}{1+y_n}$. If we would have to prove the converse we...
  14. Z

    MHB Series Convergence with Comparison Test

    Hey, I am working on Calculus III and Analysis, I really need help with this one problem. I am not even sure where to begin with this problem. I have attached my assignment to this thread and the problem I need help with is A. Thank you!
  15. D

    Convergence of Series: Finding x for Convergence | Homework Statement

    Homework Statement For which number x does the following series converge: http://puu.sh/lp50I/3de017ea9f.png Homework Equations abs(r) is less than 1 then it is convergent. r is what's inside the brackets to the power of n The Attempt at a Solution I did the question by using the stuff in...
  16. S

    Telescoping series convergence question

    Homework Statement [/B] Hello, this problem is from a well-known calc text: Σ(n=1 to ∞) 8/(n(n+2)Homework Equations [/B] What I have here is decomposingg the problem into Σ(n=1 to ∞)(8/n -(8/n+2)The Attempt at a Solution I have the series sum as equaling (8/1-8/3) + (8/2-8/4) + (8/3-8/5) +...
  17. Draconifors

    Convergence of a Complex Series

    Homework Statement "Determine whether the following series converge or diverge. If the series is geometric or telescoping, find its sum.": ## \left ( \sum_{k=1}^\infty2^{3k} *3^{1-2k} \right)## Homework Equations [/B] The different tests for convergence? The Attempt at a Solution Ok...
  18. Z

    Mathematica and Infinite Series Convergence Tests

    Hey everyone, I'm currently in Calc 2 and the only thing I seem to be having a problem with is a couple of the convergence tests. When I take pretty much any math course, I always use mathematica to help check my answers when I'm doing HW or practicing so I don't waste time. My question is...
  19. titasB

    Infinite Series Convergence using Comparison Test

    Homework Statement Determine whether the series is converging or diverging Homework Equations ∞ ∑ 1 / (3n +cos2(n)) n=1The Attempt at a Solution I used The Comparison Test, I'm just not sure I'm right. Here's what I've got: The dominant term in the denominator is is 3n and cos2(n)...
  20. C

    Quick question about Ratio Test for Series Convergence

    Homework Statement [/B] This is the question I have (from a worksheet that is a practice for a quiz). Its a conceptual question (I guess). I understand how to solve ratio test problems. "Is this test only sufficient, or is it an exact criterion for convergence?" Homework Equations Recall the...
  21. N

    Complex Analysis: Series Convergence

    Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...
  22. C

    Convergence or Divergence of a series

    Homework Statement Does sum from n=1 to n=infinity of 1/[n^(1+1/n)] converge or diverge. Homework Equations ^^^^^^^^^^^^^^^ The Attempt at a Solution The general term goes to 0 and its a p-series with p>1, but for large n the series becomes 1/n pretty much so, even tho p>1 is it divergent?
  23. Ignis Radiis

    Series Convergence and Divergence test

    Homework Statement So my question was Sum- (n=2) ln(n)/n Homework Equations I noticed that you can only limit comparison, because so far, I have tried doing all the other test such as the nth term test, p-series, integral(i have no idea how to integrate that). The Attempt at a Solution
  24. ironman

    Does This Sequence Converge or Diverge?

    Homework Statement Show if this sequence (with n=1 to infinity) diverge or converge Homework Equations [/B] The Attempt at a Solution If I use the Limit Comparison Test: compare with so you get that equals lim n -> inf => inf. Can I use the Test like this? What does this...
  25. E

    Series Convergence: Explaining P>1 & P>0

    Homework Statement Hi, everyone. I'd appreciate it if someone could explain something for me regarding the convergence of series. Thanks in advance![/B] Homework Equations In my calculus book, I'm given the following: (1) - For p > 1, the sum from n=1 to infinity of n^-p converges. (2) -...
  26. I

    MHB Determine Convergence/Divergence: Series Answers

    my final is tomorrow and my instructor gave a list of questions that will be similar to the ones on the final exam and i want to see how they should be done properly. I've been working on other problems but i can't get past these ones. thanks determine convergence/divergence...
  27. Chacabucogod

    Fourier Series Convergence Criterion

    I'm currently reading Tolstov's "Fourier Series" and in page 58 he talks about a criterion for the convergence of a Fourier series. Tolstov States: " If for every continuous function F(x) on [a,b] and any number ε>0 there exists a linear combination σ_n(x)=γ_0ψ_0+γ_1ψ_1+...+γ_nψ_n for which...
  28. A

    MHB Series Convergence Test Questions

    Just a few quick questions this time: I'm doubting the first one mostly, because when I used the integral test to evaluate it: I ended up getting (-1/x)(lnx +1) from 2 to infinity, which gave me an odd expression: (-1/infinity)(infinity +1 -ln2 -1). I'm assuming this means it is convergent...
  29. A

    MHB Series Convergence and Divergence III

    Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm highly doubtful of my answer for c. I used the roots test instead of the ratio test, which gives 1/n, which I took the limit of to get an interval of [-∞ , ∞] As for a and b, I got [-5,5] and (-∞, ∞)...
  30. A

    MHB Series Convergence and Divergence II

    Hey guys, I have a few more questions for the problem set I'm working on at the moment: I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute...
  31. A

    MHB Series Convergence and Divergence I

    Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm mostly unsure of my response for b. For a, I just split the series into two parts and added 6+3 to get 9, and thus the series is convergent. For c, I got 3/5 after taking the limit, which is...
  32. karush

    MHB Series Convergence: Test for x Values

    (x-1)-\frac{(x-1)^2}{2!}+\frac{(x-1)^3}{3!}-\frac{(x-1)^4}{4!}+ ∙ ∙ ∙ well this looks like an alternating-series, the question is: at what value(s) of x does this converge. one observation is that if x=0 then all terms are 0 so there is no convergence, also I presume you can rewrite this as...
  33. G

    Series convergence / divergence

    Homework Statement Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally? \sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n}) Homework Equations Alternating series test The Attempt at a Solution I started out by taking the limit of ##a_n...
  34. Y

    Series convergence for certain values of p

    Homework Statement For which integer values of p does the following series converge: \sum_{n=|p|}^{∞}{2^{pn} (n+p)! \over(n+p)^n} Homework Equations The Attempt at a Solution I'm trying to apply the generalised ratio test but get down to this stage where I'm not sure what...
  35. M

    Convergence of Series: Check a) & b)

    Homework Statement Check if the series below converge. a) $$\sum_{n = 1}^\infty \frac{n}{2n^2 - 1}$$ b) $$\sum_{n = 2}^\infty (-1)^n \frac{2n}{n^2 - 1}$$ Homework Equations The Attempt at a Solution For a). The series converge if the sum comes up to a finite value. If...
  36. A

    How Do You Choose Comparison Limits in Series Convergence Tests?

    I am currently learning series and testing for convergence. For comparison tests especially I am having an issue grasping the concept of picking a proper limit to compare too. For example the following problem If someone could please put it in the form where it actually looks like what it...
  37. A

    Proving Series Convergence: \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}

    Homework Statement Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges. The Attempt at a Solution I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!
  38. M

    MHB Series Convergence: Showing Convergence & Sum Equivalence

    a) Show that sum_(n=0)^infinity (2^n x^n)/((1+x^2)^n) converges for all x in R\{-1,1} b) Even though this is not a power series show that sum above = 1 + sum_(n=1)^infinity (2nx^n) for all -1<x<=1. For part a by the ratio and root test we get |(2x)/(1+x^2)| but this does not have an n in it...
  39. M

    Question on fourier series convergence

    hey pf! if we have a piecewise-smooth function ##f(x)## and we create a Fourier series ##f_n(x)## for it, will our Fourier series always have the 9% overshoot (gibbs phenomenon), and thus ##\lim_{n \rightarrow \infty} f_n(x) \neq f(x)##? thanks!
  40. O

    Series Convergence: An=Ʃ(k)/[(n^2)+k] - Find Value

    Homework Statement An=Ʃ(k)/[(n^2)+k] the sum is k=0 to n, the question is, to which value does the this series converge to Homework Equations i know for sure that this series converges, but could not figure out the value to whch it converges The Attempt at a Solution i did the...
  41. A

    Determine the values of x for series convergence

    Homework Statement Determine the values of x for which the following series converges. Remember to test the end points of the interval of convergence. ^{∞}_{n=0}\sum\frac{(1-)^{n+1}(x+4)^{n}}{n} Homework Equations I worked it down to |x+4|<1 ∴-5<x<-3 The Attempt at a Solution...
  42. L

    Does the series converge? Exploring the convergence of ln(1+e^-n)/n

    Homework Statement So I need to determine if the series \Sigmaln(1+e^{-n})/n converges.Homework Equations The Attempt at a Solution I know it does, but cannot prove it. Wolfram says that the ratio test indicates that the series converges, but when I try to solve the limit I get that it equals...
  43. 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...
  44. A

    Does Using Maximum Coefficients Determine the Smallest Radius of Convergence?

    Homework Statement Let Ʃanx^n and Ʃbnx^n be two power series and let A and B be their converging radii. define dn=max(lanl,lcnl) and consider the series Ʃdnx^n. Show that the convergence radius of this series D, is D=min(A,B) Homework Equations My idea is to use that the series...
  45. A

    Determining Convergence of Series Using Comparison and Ratio Tests

    Homework Statement Does the series \Big( \sum_{n=1}^\infty\frac{1}{(3^n)*(sqrtn)} \Big) Converge or Diverge? By what test?Homework Equations 1/n^p If p<1 or p=1, the series diverges. If p>1, the series converges. If bn > an and bn converges, then an also converges. The Attempt at a...
  46. T

    MHB Infinite series convergence III

    Test these for convergence. 5. infinity E...((n!)^2((2n)!)^2)/((n^2 + 2n)!(n + 1)!) n = 0 6. infinity E...(1 - e ^ -((n^2 + 3n))/n)/(n^2) n = 3 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?
  47. T

    MHB Infinite series convergence II

    Test these for convergence. 3. infinity E...((-1)^n)*(n^3 + 3n)/((n^2) + 7n) n = 2 4. infinity E...ln(n^3)/n^2 n = 2 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?
  48. T

    MHB Do These Infinite Series Converge?

    Test these for convergence. 1. infinity E...n!/(n! + 3^n) n = 0 2. infinity E...(n - (1/n))^-n n = 1 Btw, E means sum. Which tests should I use to solve these?
  49. C

    MHB Ratio Test Questions/ Series Convergence

    I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.
  50. P

    Series convergence vs. divergence

    Simple question: Are there any series which we don't know whether or not they converge?