Using Cauchy criterion to prove that sequence $x_n = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} +$...$+ (-1)^{n-1}\frac{1}{n}$ is convergent.

Question

I'm stuck at $$|a_{n+p} - a_n| \leq \frac{1}{n+1} + \frac{1}{n+2} + ...+\frac{1}{n+p}$$,where $\forall p \in\mathbb{N}$. How can I find a N where $\forall n > N, |a_{n+p} - a_n| < \epsilon$ ?

Answer 1

You need to modify the series expression with $b_n=a_{2n}-a_{2n+1}=\frac{1}{2n}-\frac{1}{2n+1}=\frac{1}{2n(2n+1)}$.
Use Cauchy criterion on $\sum b_n$.