Given the alternating harmonic series: $$ \sum^{\infty}_{n=1}{\frac{(-1)^{n+1}}{n}} $$
and if we let the sequence of its partial sums be $s_n$ then how can we express the sequence of even partial sums $s_{2n}$ and odd partial sums $s_{2n+1}$?
I did the following but am not completely sure if it is valid:
$$ s_{2n}= \sum^{2n}_{i=1}{\frac{(-1)^{i+1}}{i}}$$
$$ s_{2n+1}= \sum^{2n+1}_{i=1}{\frac{(-1)^{i+1}}{i}}$$
Then to show the first is increasing and the second is decreasing, can we just take $s_{2n+2}-s_{2n}$ and show that this is $> 0$? Similarly then taking $s_{2n+3}-s_{2n+1}$ and showing that this is $< 0$?
And to go one step further, how could we show that each of these sequences of partial sums has a limit? And then show that the limits are the same?
Best Answer
What you have done is fine. Showing the increasing/decreasing nature works well. You can invoke the alternating series theorem to show the whole series has a limit because the terms are alternating in sign and decreasing monotonically to zero. If the whole series has a limit, every subseries converges to the same limit and you are done.