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Does the series of 1/n^(1+a) converge or diverge

  1. Jan 3, 2009 #1
    1. The problem statement, all variables and given/known data

    I have to determine whether the infinite sum (from n=1) of 1/n^(1+a) converges or diverges. Where 0 < a < 1

    2. Relevant equations

    Ration Test

    Comparison / Integral test

    3. The attempt at a solution

    I have tried using the ratio test but it gives me 1, so it cannot determine convergence or divergence.

    I then tried using comparision test, comparing it to 1/n (which diverges) and 1/n^2 (which converges). But since 1/n^(1+a) is smaller than 1/n and greater than 1/n^2 I cannot make such comparison.

    Then I used integral test where I let a=0.1 . This method tells me that for (a) greater than or equal to 0.1 the series converges. But how do I show that it converges (I think it converges) for (a) less than 0.1?

  2. jcsd
  3. Jan 3, 2009 #2


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    Staff Emeritus
    Science Advisor

    Re: convergence/divergence

    Why "a= 0.1"? Why not just leave it as "a"?
  4. Jan 3, 2009 #3
    Re: convergence/divergence

    Hi thanks for the reply.

    Leaving it as (a), I got the answer as

    (1/(-2-a)).((1/n^(2+a)) - 1)

    If this is right, then it will converge to 1.

    So am does this mean that the series will converge to 1 as well?

    thanks again
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