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

[v_pino]

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

 

[HallsofIvy]

Why "a= 0.1"? Why not just leave it as "a"?

 

[v_pino]

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