Series convergence Definition and 111 Threads

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




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

    Series Converg. Hmwk: Determine Convergence of \sum(-1)^n n/(n^p + (-1)^n)

    Homework Statement Really tough series to work with. Determine the convergence ( absolute or conditional ) or divergence of : ##\sum_{n=2}^{∞} \frac{(-1)^n n}{n^p + (-1)^n}## Homework Equations ?? Series tests? The Attempt at a Solution This series is really ugly. I'm not sure how to...
  2. S

    So, the infinite series converges for a>2 and diverges for a=2.

    Homework Statement Show that the infinite series \sum_{n=0}^{\infty} (\sqrt{n^a+1}-\sqrt{n^a}) Converges for a>2 and diverges for x =2The Attempt at a SolutionI'm reviewing series, which I studied a certain time ago and picking some questions at random, I can't solve this one. I tried every...
  3. L

    MHB How to Analyze Series Convergence with a Floor Function?

    I have one series \sum_{n=13}^{\infty}(-1)^{\left\lfloor\frac{n}{13}\right\rfloor} \frac{ \ln(n) }{n \ln(\ln(n)) } . How to investigate its convergence? I wanted to group the terms of this series but I don't know whether it's a good idea as we have 13 terms with minus and then 13 with plus and...
  4. C

    QED perturbation series convergence versus exact solutions

    It is well known due to the famous argument by Dyson that the perturbation series for quantum electrodynamics has zero radius of convergence. Dysons argument essentially goes like that: If the power series in α had a finite (r>0) radius of convergence it also would converge for some small...
  5. L

    MHB Solving Series Convergence Problems: 1+ \frac{1}{3}-\frac{1}{2}+\frac{1}{5}+...

    I have a problem with convergence of two series: 1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+... 1+ \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{1}{...
  6. B

    Does the series \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) converge?

    Homework Statement Determine whether the following series diverges, converges conditionally, or converges absolutely. \sum^{\infty}_{n=1}sin(\frac{1}{n^{4}}) Homework Equations The Attempt at a Solution This was on today's test, and was the only problem I wasn't able to solve. I doubt my...
  7. U

    Testing for series convergence.

    Homework Statement \sum(\frac{2n}{2n+1})n2 (The sum being from n=1 to ∞). Homework Equations The Attempt at a Solution Used exponent properties to get (\frac{2n}{2n+1})2n. Using the root test, the nth root of an = lim n->∞(\frac{2n}{2n+1})2 = 1. However, the root test is...
  8. B

    Another question about Fourier series convergence

    I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense. Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if...
  9. M

    Evaluate Series Convergence Analytically

    Hi, while reading some artificial intelligence book, i came upon the following sum. How can I evaluate it analytically, so not guess it by computing many terms? It's easy to see by ratio test that it converges (intuitively too, since its a linear vs exponential function). \sum_{i=1}^\infty...
  10. C

    Infinite series convergence question:

    Homework Statement Does \sum _{ n=1 }^{ \infty }{ \frac { { \alpha }^{ n }{ n }! }{ { n }^{ n } } } converge \forall |\alpha |<e and if so, how can I prove it? Homework Equations { e }^{ x }=\sum _{ n=0 }^{ \infty }{ \frac { { x }^{ n } }{ n! } } The Attempt at a Solution...
  11. R

    Fourier Series Convergence at the Origin

    Homework Statement The Attempt at a Solution Obviously brackets mean something other than parentheses because .5[0 + 0] ≠ .5
  12. R

    Understanding Fourier Series Convergence: Common Confusions Addressed

    Homework Statement This series is what dictates the graph above. The Attempt at a Solution I don't understand what's going on. If they're using the series that i pasted below then why aren't they multiply each value in the brackets by -2/pi? I also don't get why terms...
  13. G

    Finding the Annulus of Convergence for a Laurent Series

    I am trying to understand the idea of annulus of convergence. This is the example I have been looking at but it has me completely stumped. [∞]\sum[/n=1] (z^n!)(1-sin(1/2n))^(n+1)! + [∞]\sum[/n=1] (2n)!/[((n!)^2)(z^3n)] All of the examples I have worked on in the past have been...
  14. M

    Calculus 2, Series Convergence Questions?

    I have some problems here with Series and Convergence... Here are the problems and my guesses at it. http://img822.imageshack.us/img822/9523/23341530.png It won't tell me which one is wrong, but it just says one/all is wrong. Any help is appreciated. Attempts at solving, I tried...
  15. 1

    I want to know more about series convergence (elementary)

    I am able to use a variety of methods to check to see if a series converges, and I can do it well. However, it's not something I feel like I've intuitively conquered. I don't understand why the series 1/x diverges. I mean, I do, in that I know the integral test will give me the limit as x ->...
  16. T

    Difficult series convergence proof

    Homework Statement Show that given some ε > 0, there exists a natural number M such that for all n ≥ M, (a^n)/n! < ε Homework Equations The Attempt at a Solution Ok so I know this seems similar to a Cauchy sequence problem but its not quite the same. So I am looking for a...
  17. L

    Series Convergence and Divergence

    Homework Statement Determine if the following series converges or diverges. If it converges determine its sum. Ʃ1/(i2-1) where the upper limit is n and the index i=2 Homework Equations The General Formula for the partial sum was given: Sn=Ʃ1/(i2-1)=3/4-1/(2n)-1/(2(n+1) The...
  18. T

    Complex Power Series Convergence Help

    Homework Statement I have a problem set that asks me to determine, first, the radius of convergence of a complex series (using the limit of the coefficients), and second, whether or not the series converges anywhere on the radius of convergence. Homework Equations As an example: Σ(z+3)k2 with...
  19. H

    Fourier series convergence - holder continuity and differentiability

    Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...
  20. B

    Determine Series Convergence Given Convergence of a Power Series

    Homework Statement I am asked to comment on the convergence/divergence of three series based on some given information about a power series: \sum_{n=0}^{\infty}c_nx^n converges at x=-4 and diverges x=6. I won't ask for help on all of the series, so here's the first one...
  21. B

    Determining Series Convergence using the Ratio Test

    Homework Statement I'm asked to specifically use the Ratio Test (formula below) to determine whether this series converges or diverges (if it converges, the value to which it converges is not needed.) \sum_{n=1}^{\infty}\frac{n}{(e^n)^2} Homework Equations Ratio Test: If a_n is a sequence...
  22. 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) -...
  23. J

    Fourier series convergence question

    Homework Statement f(x) = 5, -pi <= x <= 0 f(x) = 3, 0 < x <= pi f(x) is the function of interest Find the x-points where F(x) fails to converge to f(x) Homework Equations F(x) = f(x) if f is continuous at x\in(-L,L) F(x) = 0.5[ f(x-) + f(x+) ] if f is...
  24. Telemachus

    Radius of Convergence for Power Series: What is the Limiting Ratio Test?

    Homework Statement Hi there. Well, I was trying to determine the radius and interval of convergence for this power series: \displaystyle\sum_{0}^{\infty} \displaystyle\frac{x^n}{n-2} So this is what I did till now: \displaystyle\lim_{n \to{+}\infty}{\left...
  25. R

    Interval of Convergence: Series f(x)

    Homework Statement A function f is defined by... f(x) = \frac{n+1}{3^{n+1}} x^n a.) find the interval of convergence of the given power series. b.) Find \lim_{x\rightarrow 0} \frac{f(x) - \frac{1}{3}}{x} c.) Write the first three nonzero terms and the general term for...
  26. S

    Series Convergence: Does \sum_{k=1}^{\infty }{a_k}^{5/4} Necessarily Converge?

    Homework Statement Say that \sum_{k=1}^{\infty }a_k converges and has positive terms. Does the following necessarily converge? \sum_{k=1}^{\infty }{a_k}^{5/4} Homework Equations If it necessarily converges, a proof is required, if not, a counter-example is required. The...
  27. H

    Question: Series Convergence for ((-1)^n*n!)/(1*6*11*...*(5n+1))

    Homework Statement Does the series ((-1)^n*n!)/(1*6*11*...*(5n+1)) from n = 0 to \infty absolutely converge, converge conditionally or diverge? Homework Equations The Attempt at a Solution I did the ratio test for ((-1)^n *n!)/(5n+1)) and I found that it diverges but apparently...
  28. O

    Proof About Series Convergence

    Homework Statement let an be a positive series. it is known that for every bn\rightarrow \infty the sum from 1 to inf of an/bn is convergent prove that the sum from 1 to inf of an is convergent Homework Equations The Attempt at a Solution I thoght maybe to try to say. let...
  29. Telemachus

    Convergence of Series: Comparing Criteria & Quotient Limit

    Homework Statement Well, hi there. I have to study the convergence of the next series using the comparison criteria, or the comparison criteria through the limit of the quotient. \displaystyle\sum_{n=1}^\infty{\displaystyle\frac{3n^2+5n}{2^n(n^2+1)}} I think that I should use the...
  30. J

    Series convergence- why is my proof incorrect?

    Homework Statement Prove that if \sum{|a_{n}|} converges and (b_{n}) is a bounded sequence, then \sum a_{n}b_{n} converges.Homework Equations Comparison Test part (i): Let \sum a_{n} be a series where a_{n}\geq 0 for all n. If \sum a_{n} converges and |b_{n}|\leq a_{n} for all n, then \sum...
  31. B

    Is Power Series Convergence Related to Other Series Convergence?

    Homework Statement If \sum_{n=0}^{\infty} c_{n}4^n is convergent, does it follow that the following series are convergent? a) \sum_{n=0}^{\infty} c_{n}(-2)^n b) \sum_{n=0}^{\infty} c_{n}(-4)^n Homework Equations The Power Series: \sum_{n=0}^{\infty} c_{n}(x - a)^n The...
  32. W

    Convergence of Power Series without Recursion Relation

    Homework Statement Suppose I have the power series: f(x) = A0 + A1 x +A2 x^2 ...An x^n Where A0..An are numbers, there is no recursion relation. Find the interval of convergence Homework Equations The Attempt at a Solution Can I use ratio test? How would I do this since there is no recursion...
  33. estro

    Conditional Convergence and the Comparison Test: A Proof by Contradiction

    Series convergence "by Parts Supose: \sum c_n = \sum (a_n+b_n) (*1) \sum a_n is conditionaly convergent (*2) \sum b_n is absolutly convergent (*3) And I have seen this proof: [Proving \sum c_n is conditionally convergent] From (*1) and (*2) \Rightarrow \sum c_n its...
  34. estro

    Series convergence and Cauchy criterion

    The Attempt at a Solution * forgot to state that I choose m > n > max { N_1, N_2 }. I'm not sure if i did it right, but seems ok to me =) Will appreciate your opinion...
  35. D

    Series Convergence: Can I Create a p-Series?

    Homework Statement infinity SIGMA sqrt(n) / ((n^2)(ln(n)) n = 2 Homework Equations The Attempt at a Solution Could i beat this into a p-series perhaps?
  36. M

    Ratio Test for series convergence factoring problems

    Homework Statement \Sigma2nn!/(n+2)! Homework Equations I'm using the ratio test because there are factorials but I'm a little stuck on whether or not to factor out The Attempt at a Solution lim 2n+1(n+1)!/(n+3)!*(n+2)!/2n(n)! After I set it up here I'm not sure of how to factor...
  37. F

    Convergence of Infinite Series: Two Examples

    Okay, there's two questions, actually. First, determine if the series converges. SUM: (n-2)/(n^2-4n) (from n=5 to infinity) I used the integral test, found the integral to be 1/2 log(n^2-4n) from x=5 to x=t as t approaches infinity. That turned out to go to infinity so the series...
  38. Somefantastik

    Series Convergence: Does This Series Converge and What is the Sum?

    Homework Statement Does it converge, and what is the sum: \sum_{n=1}^{\infty}\frac{1}{n n^{\frac{1}{n}}} Homework Equations The Attempt at a Solution Please check my method and conclusion: Using the root test: \displaystyle\lim_{n\to\infty}\left|\frac{1}{n...
  39. H

    Series Convergence: Show AK -> 0 as K->∞

    Homework Statement Show that if \sumak converges, then \sum from k to ∞ of ak goes to zero as k goes to ∞. Homework Equations The Attempt at a Solution I'm not really sure how to go about this proof. But, this is my attempt, First I tried to show that \sumak is convergent. Let c be a...
  40. P

    Is Sigma(y(n)) Absolutely Convergent if y(n) = O(x(n))?

    Hi, assume sigma(x(n)) is an absolutly convergent series and that y(n)=O(x(n)), then can we conclude that sigma(y(n)) is absolutly convergent? thanks
  41. D

    Convergence of Squared Series Proof for Positive Sequences

    Homework Statement Assume that the series(an) is convergent and that an >= 0 for all n in N. Prove that the series((a^2)n) converges. Homework Equations The Attempt at a Solution Alright, this is what I've got so far: Assume that the series of an is convergent and that an>=0...
  42. S

    Convergence of Series with Square Root Terms

    Hi all, A friend of mine asked me if i had any ideas about the following problem, i tackled it but with no success, so i thought i would post it here. It is not a homework problem, or a regular textbook problem. Problem: If we know that a series with positive terms ...
  43. T

    Is the Series \(\sum_{i=1}^{\infty} \ln(\cos(\frac{1}{n}))\) Convergent?

    Homework Statement Check if the following series is convergent. \sum^{\infty}_{i=1}l n(cos(\frac{1}{n})) I have tried a lot of different tests without success. I need some hint. Thanks Homework Equations The Attempt at a Solution
  44. U

    Convergence Series: Partial Fractions Homework

    Homework Statement Use partial fractions to show \displaystyle\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)} = \frac{1}{4} The Attempt at a Solution I did the partial fraction decomposition to get: \displaystyle\sum_{n=1}^\infty \frac{1}{2n} - \frac{1}{n + 1} + \frac{1}{2n + 4} I'm not...
  45. camilus

    Can the value of a be solved for in the infinite series convergence equation?

    \sum^{\infty}_{x=1} \frac{cos(14.1347 \ln (x))}{x^{a}} = 0 Is there a way to solve for a? I don't think so but maybe someone here will have an insight as to what to do..
  46. T

    Alternating series convergence

    Homework Statement I have 2 McLaurin series. 1) ln(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... 2) ln (1-x) = -x - (x^2)/2 - (x^3)/3 - (x^4)/4 + ... The Attempt at a Solution I want to find the range of x values for which series 1) and 2) converge. For 1) I am using the...
  47. T

    ODE - Power Series Convergence

    Homework Statement Solve (1-4x^2)y''+34x\cdot y'-70y=0 Homework Equations Basically, I found the recurrence relationship to be: a_{n+2}=\frac{2 (-7 + n) (-5 + 2 n)}{(n+1)(n+2)}a_n} Now, I solve for y1 where y1 had a_0=0 and a_1 = 1. It is a simple polynomial of degree 7...
  48. 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 =...
  49. M

    What are the conditions for series convergence in terms of \Gamma_{2n+1}?

    what are the conditions on \Gamma_{2n+1} so as for the series to converge ? m\pi=\sum^{\infty}_{n=0} \frac{(-1^{n}) \Gamma_{2n+1}}{(2n+1) r^{2n+1}} m = 0,1,2,... r is real number is there an explicit expression for \Gamma_{2n+1} in terms of n ?
  50. S

    Do Merged Mathematical Sequences Converge to the Same Limit?

    thres two series An and Bn , Bn unite with An from some index. need to prove that limAn_k = limBn_m Bn_m and An_m are sub series . TNX !
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