For what values of p does the alternating series 1/n^p converge?

[tarheelborn]

Homework Statement

For what values of p does the series [1/1^p - 1/2^p + 1/3^p - 1/4^p +... converge?

Homework Equations

The Attempt at a Solution

I believe that this series converges for all p \in N because the sequence of a_n's is nonincreasing and converges to 0. I am not quite sure, however, to show that it converges to 0. I know that the sequence 1/n converges to 0 and I know that p is fixed, but I don't know how to massage that information into what I need. Thanks.

 

[clamtrox]

If the series is alternating, then you only need to show that $$|a_n| \leq |a_{n-1}|$$ and that the sequence of a_ns converges to zero.

A sequence converges to zero if for any positive real number $$\epsilon$$, you can find a N large enough so that $$a_n < \epsilon$$ for all n>N.

 

[tarheelborn]

Right; I understand the epsilon proof and the theorem related to alternating series. Although I know it sounds really dumb, I am having trouble finding N.

 

[clamtrox]

$$\frac{1}{N^p} = \epsilon.$$

Now, for certain kinds of p you can always find an N for every epsilon.

 

[tarheelborn]

I was thinking for p >= 0. Thank you so much.