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. Somefantastik

    Uniform Convergence: Series Homework Help

    Homework Statement Show uniform convergence \frac{4b}{\pi} \sum^{\inf}_{n=1} \frac{1-(-1)^{n}}{n^{2}}cos(nt)cos(nx) for fixed tHomework Equations The Attempt at a Solution \left| cos(nt) \right| \leq 1 \left| cos(nx) \right| \leq 1 lim \left|\frac{1 - (-1)^{n}}{n^{2}}\right| \ = \ 0...
  2. S

    Homework: Investigating Infinite Series Convergence

    Homework Statement a) consider the infinite series (k=1) sum (inf) [(k+1)^(1/2) - (k)^(1/2)] expand and simplify the nth partial sum. determine wether the oartial sums S_n converge as n-> inf b) determine all the numbers x in R so that the infinite series (k=0) sum (inf) [x^(k)/(k!)]...
  3. W

    Solve Series Convergence: $\sum^{n=0}_{\infty}\frac{2n-1}{\sqrt{n^{5}+1}}

    [SOLVED] Series convergence Homework Statement \sum^{n=0}_{\infty}\frac{2n-1}{\sqrt{n^{5}+1}} Homework Equations The Attempt at a Solution
  4. A

    Series convergence representation

    Homework Statement \sum_{n=0}^\infty (0.5)^n * e^{-jn} converges into \frac{1}{1-0.5e^{-jn}} Prove the convergence. Homework Equations Power series, and perhaps taylor & Macclaurin representation of series. The Attempt at a Solution This isn't a homework problem...
  5. P

    Does \(\sum_{n=2}^{\infty}\frac{1}{(\ln n)^k}\) Converge for \(k > 1\)?

    It's easy to see that \sum_{n=2}^{\infty}\frac{1}{lnn} does not converge. But what happens to \sum_{n=2}^{\infty}\frac{1}{(lnn)^k} with k > 1 and why? Can anybody help?
  6. A

    Series Convergence: Trouble Determining Convergence/Divergence

    I'm having trouble determining whether these series converge or diverge. 1. sigma sqrt(n/(n^4-2)) I tried ratio test, but it gave me 1 as the answer (indeterminate) 2. sigma sin (pi/x) 3. sigma sin(x) I know that sin(x) is bounded... Any hints?
  7. B

    Convergence of Series: Is x ≤ 2 the Only Condition for Convergence of S?

    I have the following series: S = \sum _{n=0} ^{\infty} 4^n (x+2)^n Is that the same as 4^n \sum_{n=0} ^{\infty} (x+2)^n = 4^n ((x+2) + (x+3) + \cdots + (x+n)) ? Best Regards Bob
  8. C

    Proving Series Divergence: Convergence of a_n w/o Explicit Formula

    Given a_{n} > 0 and \sum a_{n} diverges, show that \sum \frac{a_{n}}{1+a_{n}} diverges. Since I don't have an explicit form for the series, I can't apply any of the standard tests. I'm not sure where to start on this problem. I know the criteria for convergence/divergence, namely tail end of...
  9. B

    Solving a Series Convergence Problem: Can You Help?

    Hi, I'm having trouble getting the sum of the following series. I'm pretty sure that it is convergent. \sum\limits_{n = 1}^\infty {\frac{{\left( { - 3} \right)^n }}{{7^n }}} Here is what I have done. \sum\limits_{n = 1}^\infty {\frac{{\left( { - 3} \right)^n }}{{7^n }}} = -...
  10. D

    Quick series convergence question

    Hey all, it's been a while since I've done series and I have a quick question. How would I show the convergence or divergence of \sum \left(\sqrt{n+1}-\sqrt{n}\right)? The ratio test is inconclusive I think, and I'm not sure how I would go about doing the root test. Or is there a series I...
  11. Y

    Series Convergence: Region in xy-Plane & Tests to Use

    Let D be the region in the xy-plane in which the series Sum k=1 to k= infinity (x+2y)^k /k converges. Then the interior of D is: The open region between two parallel lines. Can someone explain why this is true? You don't need to work out a full blown solution. What convergence tests...
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