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

1. Jan 3, 2009

### v_pino

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?

Thanks

2. Jan 3, 2009

### HallsofIvy

Staff Emeritus
Re: convergence/divergence

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

3. Jan 3, 2009

### v_pino

Re: convergence/divergence