# Series 5.

1. Nov 19, 2007

### azatkgz

1. The problem statement, all variables and given/known data
Determine whether the series converges absolutely,converges conditionally or diverges.

$$\sum_{n=1}^{\infty}\ln\left(1+\frac{(-1)^n}{n^p}\right)$$
where p is a some parameter

3. The attempt at a solution

$$\ln\left(1+\frac{(-1)^n}{n^p}\right)=\frac{(-1)^n}{n^p}-\frac{1}{n^{2p}}+\frac{(-1)^{3n}}{3n^{3p}}+O(\frac{1}{n^{4p}})$$

Here
$$\sum_{n=1}^{\infty}\frac{1}{n^{2p}}$$ converges for p>1/2

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^p}$$ converges absolutely for p>1

My answer is the series converges for p>1/2

for $$\frac{1}{2}<p\leq 1$$ it converges conditionally

for $$p>1$$ it converges absolutely