What is Convergence test: Definition and 35 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. Vividly

    B Understanding about Sequences and Series

    Homework Statement:: Tell me if a sequence or series diverges or converges Relevant Equations:: Geometric series, Telescoping series, Sequences. If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too? Also if I...
  2. S

    Determining if a series converges

    The following is my attempt at the solution. Here, I used limit comparison test to arrive at the answer that the series converges. However, the answer sheet reads that the series diverges. I am confused because I cannot figure where my work went wrong… can anyone tell me how the series...
  3. S

    Using comparison tests and limit comparison test

    The answer sheet states that the series converges by limit comparison test (the second way). In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way) Thank you!
  4. 0

    Integral of 1/ln(x). Convergence test

    Some functions have straight foward integrals, but they get complicated if you take the inverse of it. 1/f(x) for instance. The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges. ##\int_1^\infty (\ln(x))^n dx## If n = 0, I...
  5. Rectifier

    Does this series converge? Using the limit comparison test

    The problem In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## : $$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$ The attempt ## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} =...
  6. binbagsss

    Elliptic functions proof -- convergence series on lattice

    Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...
  7. P

    What is the Limit of e^(1/x) as x Approaches 0 and the Direction Matters?

    In this solution, in the last 3rd line, I get the first part (-e^-1 - e^-1), however, after the '-' symbol, the person writes (1/b * e^1/b - e^1/b) and takes the limit as b->0. However, shouldn't this give him (inf. * e^inf - e^inf)? Thanks (His answer is correct, by the way)
  8. enh89

    Why does it matter what convergence test I use?

    I just took a calc 2 test and got 3/8 points on several problems that asked you to show convergence or divergence. The reason being that I didn't use the correct test of convergence? The answer was right, if you get to the point where you know the series converges, then why does it matter which...
  9. B

    Help with convergence test?

    Which of these converge? 1. 5n/ (2n-1) 2. e^n/ n 3. e^n/ (1+e^n) Attempt: 1) lim n-> ∞ 5n/(2n-1) = 5n/2n = 5/2 ≠ 0 so diverge? 2) Change n to x e^x/ (1 + e^x) Antiderivative: ln |1 + e^x| lim t->∞ of ln |1 + e^x| ln |1 + e^∞| - ln |1 + e^0| Got stuck here 3) Help :(
  10. B

    VERY CONFUSED with convergence test?

    n^2 - 1 / (n^3 + 6n) If I use the nth divergence test, I plug ∞ in (limit as n -> ∞) for n and since the degree on the bottom is larger I get 0, which means it converges. However, if I use the limit comparison test and compare it to: n^2/n^3, which = 1/n, which diverges -> n^2 - 1 / (n^3 + 6n)...
  11. arpon

    Convergence Test: P(z) = 1 - z/2 + z^2/4 - z^3/8

    Homework Statement ##P(z) = 1 - \frac{z}{2} + \frac{z^2}{4} - \frac{z^3}{8} + ... ## Determine if the series is convergent or divergent if ## |z| = 2 ##, where, ## z## is a complex number. Homework Equations ##1+r+r^2+r^3+...+r^{N-1}=\frac{1-r^N}{1-r}## The Attempt at a Solution Let, ##z = 2...
  12. ognik

    MHB Please check this convergence test (#2)

    $ \sum_{n}\frac{1}{n.{n}^{\frac{1}{n}}} $ Now $\frac{1}{n}$ diverges and $\ne 0$ , so by limit comparison test: $ \lim_{{n}\to{\infty}} \frac{n.{n}^{\frac{1}{n}}}{n} = \lim_{{n}\to{\infty}} {n}^{\frac{1}{n}} = \lim_{{n}\to{\infty}} {n}^0 = 1$ (I think the 2nd last step may be dubious?)...
  13. ognik

    MHB Please check this convergence test

    $ \sum_{n} \ln\left({1+\frac{1}{n}}\right) $ $ \ln\left({1+\frac{1}{n}}\right) = \ln\left({1}\right) + \ln\left({\frac{1}{n}}\right) = 0 +\ln\left({{n}^{-1}}\right) = -\ln\left({n}\right)$ Now $\lim_{{n}\to{\infty}} -\ln\left({n}\right) \ne 0$, therefore the series diverges. (Also can you...

    Converging Series: Comparison Test w/ 1/n^2

    Mod note: Moved from Homework section I know that ##1/n^4## converges because of comparison test with ##1/n^2## (larger series) converges . how do I know ##1/n^2## converges? coz I cannot compare it with ##1/n## harmonic series as it diverges. @REVIANNA, if you post in the Homework & Coursework...
  15. ognik

    MHB How to Apply Gauss's Convergence Test to a Hypergeometric Series?

    I found the interval of convergence for a hypergeometric series as |x| < 1, now I believe that I need to apply 'Gauss's test' to check the end point(s). For $ \left| x \right|=1 $ my $ \left| \frac{{a}_{n}}{{a}_{n+1}} \right| = \left|...
  16. J

    Doubt about convergence test on differential equations

    I will try to explain this with an analogy. Let's have this equation: x^2 =9 And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put...
  17. H

    Raabe's test says Legendre functions always converge?

    The Legendre functions are the solutions to the Legendre differential equation. They are given as a power series by the recursive formula (link [1] given below): ##\begin{align}y(x)=\sum_{n=0}^\infty a_n x^n\end{align}## ##\begin{align}a_{n+2}=-\frac{(l+n+1)(l-n)}{(n+1)(n+2)}a_n\end{align}##...
  18. 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...
  19. I

    Convergence Test Problem

    Homework Statement determine whether the Ʃ n4 / en2 is convergent or divergent? Homework Equations The Attempt at a Solution Using Root test: lim of n4/n / en as n approaches infinity But lim of n4/n as n approaches infinity = ∞0 So: Let N = lim of n4/n as n approaches...
  20. I

    Convergence Test for Alternating Series and When to Use It

    Homework Statement Ʃ cos(k*pi)/k from 1 to infinity. This is a test for convergence. and when is the proper time to use the alternating series test like using it on (-1)k(4k/8k) would result to divergence since lim of (4k/8k) is infinity and not 0 but the function is really convergent...
  21. 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...
  22. O

    Convergence Test (Comparison) Questions

    Hello everyone, I need some help on doing convergence tests (comparisons I believe) on some Ʃ sums. I have three, they are: 1. Ʃ [ln(n)/n^2] from n=1 to ∞. I tried the integral test but was solved to be invalid (that is, cannot divide by infinity). Therefore I believe it to be a...
  23. P

    Convergence Test for Series with Cosine Cubed Terms

    Hi, How may I know whether the series ((-1)^n)[cos (3^n)x]^3/(3^n) converges/diverges?Should I use the Leibniz Criterion? It is stated that (cos a)^3 = (1/4)(3cos a + cos 3a)
  24. J

    Fourier series convergence test

    Homework Statement A function f(x) is given as follows f(x) = 0, , -pi <= x <= pi/2 f(x) = x -pi/2 , pi/2 < x <= pi determine if it's Fourier series (given below) F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx) -...
  25. P

    Help Convergence test for series

    Find the value of x so that the series below converge.[/b] \sum 1/ [(k^x) * (2^k)] (k=1 to \infty) Using ratio test, I 've got [(1/2) * 1^x] < 1 for all x in R But when I use different value of x, series converge and diverge! Really need your help! Thank you so much
  26. estro

    Dirichlet's Convergence Test - Improper Integrals

    Hello, I have question about using Dirichlet's Convergence Test which states: 1. if f(x) is monotonic decreasing and \lim_{x\rightarrow \infty} f(x)=0 2. G(x)=\int_a^x g(t)dt is bounded. Then \int_a^\infty f(x)g(x)dx is convergent. But what about the following situation: f(x)=1/x g(x)=cosxsinx...
  27. L

    Convergence Test: Solving Homework on (n!)/(2n)!

    Homework Statement It's from sum (n=1, to infinity.. I apologize for not knowing how to type it in properly!) of (n!)/(2n)! Homework Equations The Attempt at a Solution We're supposed to use either the Root Test or the Ratio Test to determine if the series converges or not. My...
  28. R

    Proving Convergence Test: 5 Statements Explained

    There are five statements: (a) If n^2 an → 0 as n → ∞ then ∑ an converges. (b) If n an→ 0 as n → ∞ then ∑an converges. (c) If ∑an converges, then ∑((an )^2)converges. (d) If ∑ an converges absolutely, then ∑((an )^2) converges. (e) If ∑an converges absolutely, then |an | < 1/n for all...
  29. M

    Which Convergence Test to Use for ∑(k(k+2))/(k+3)^2 in Calculus?

    Homework Statement ∑(k(k+2))/(k+3)^2 I honestly have no idea where to start with this I was going to try a ratio test but I wasn't sure if it would be (k+1)(k+3)/(k+4)2*(k+3)2/(k(k+2) Homework Equations [b]3. The Attempt at a Solution
  30. R

    D'Alembert Ratio Test: Convergence test

    Hi this is just a general question about using the ratio test for convergence. If I have to carry out the test to find out if something converges (and I don't need to find out if its absolutely converges, but just convergence), then can my answer to the test be negative? Or does the...
  31. J

    Factorial Series Convergence: Investigating the Sum of n!/10^n

    Homework Statement Show that \sum \frac{n!}{10^n} converges or diverges.(Note, I was unsure of how to format this via latex, so the summation is from n = 1 to infinity.)Homework Equations The root test: |\frac{a_n_+_1}{a_n}| The Attempt at a Solution a_n=\frac{n!}{10^n}...
  32. W

    Convergence Test for Series with Exponential and Polynomial Terms

    I am having problems with the following question: Using an appropriate convergence test, find the values of x \in R for which the following series is convergent: (\sumnk=1 1/ekkx)n I used the ratio test to solve this but I'm not so sure about my solution: n1 = \frac{1}{e} n2 =...
  33. rocomath

    Convergence Test: \sum_{n=1}^{\infty}\frac{(-1)^{n}n}{n^{2}+25}

    \sum_{n=1}^{\infty}\frac{(-1)^{n}n}{n^{2}+25} Ratio Test \lim_{n\rightarrow\infty}|\frac{(-1)^{n+1}(n+1)(n^{2}+25)}{[(n+1)^{2}+25](-1)^{n}n}| \lim_{n\rightarrow\infty}|\frac{n^{3}+n^{2}+25n+25}{ n^{3}+2n^{2}+26n}|=1 Thus, the Ratio Test is inconclusive. So what should my next step...
  34. R

    Convergence Test for 1/(sqrt(k+5)) - How to Determine Series Convergence?

    Homework Statement Determine whether the series converges: 1/(sqrt(k+5)) Homework Equations well i think I'm supposed to use the integral test, which has you take the integral of the series and the limit too.. The Attempt at a Solution .. I'm just not sure how to begin, for some...
  35. S

    Very Quick Question: Which Convergence Test to Use?

    What is the best convergence test to use for the sum from n=2 to infinity of ln(n)/n^2? The comparison test and limit comparison test both probably work... but what is the right comparison for each of these tests? I have always had a hard time deciding which tests to use, especially when the...