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

  • Thread starter v_pino
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  • #1
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Homework Statement



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


Homework Equations



Ration Test

Comparison / Integral test


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?

Thanks
 

Answers and Replies

  • #2
HallsofIvy
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Why "a= 0.1"? Why not just leave it as "a"?
 
  • #3
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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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