Say I have: $(x_n)$ a sequence of real numbers such that $\sum x_n$ which converges conditionally and implies $\sum x_{2n}$ diverges.
I want to show that $x_{2n}$ does not in general converge.
So I have picked out a candidate for $\sum x_n$ = $\sum \frac{-1^{n+1}}{n}$
and so $\sum x_{2n} = \sum \frac{-1^{2n+1}}{2n}$
How can I argue rigorously that the first series converges and the second one diverges? I am aware that these both share the properties.
For the first series, I used an alternating series test,
since if we let $(b_n) = \frac{1}/{n}$ Then we see that $b_1 \geq b_2 \geq b_3 \dots $
and also that $(b_n) \to 0$ So then $ \sum (-1)^{n+1} b_n $ converges.
But how do I show divergence for the second series?