Math Help - Summation of (-1)^n/(n+(-1)^n+1)

1. Summation of (-1)^n/(n+(-1)^n+1)

Does the series in the attached image converge?

This is my attempt at a solution:

We have:
-1/2 + 1 -1/4 + 1/3 -1/6 + 1/5 -.....

I had this in an exam paper and answered that the series converges because the series:
1 - 1/2 + 1/3 - 1/4 + ...
converges (according to Liebniz).

But I only got 2/10 for the question because rearranging an infinite series can change the result. So how do I prove this series does or does not converge?

Thanks

2. Re: Summation of (-1)^n/(n+(-1)^n+1)

Originally Posted by durrrrrrrr
Does the series in the attached image converge?

This is my attempt at a solution:

We have:
-1/2 + 1 -1/4 + 1/3 -1/6 + 1/5 -.....

I had this in an exam paper and answered that the series converges because the series:
1 - 1/2 + 1/3 - 1/4 + ...
converges (according to Liebniz).

But I only got 2/10 for the question because rearranging an infinite series can change the result. So how do I prove this series does or does not converge?
Let $S_n$ denote the sum of the first n terms of the series. Then $S_{2n}$ is the same as the sum of the first 2n terms of the Leibniz series (because there is no problem with rearranging the terms in a finite sum). So $S_{2n}$ converges to the same limit L as the Leibniz series.

Next, the (2n+1)th term in the series is $-\tfrac1{2(n+1)}$, which converges to 0 as $n\to\infty.$ Therefore $S_{2n+1} = S_{2n} -\tfrac1{2(n+1)} \to L+0 = L.$

Finally, since $S_{2n}$ and $S_{2n+1}$ converge to the same limit L, it follows that the whole series converges to L.

3. Re: Summation of (-1)^n/(n+(-1)^n+1)

Originally Posted by Opalg
Finally, since $S_{2n}$ and $S_{2n+1}$ converge to the same limit L, it follows that the whole series converges to L.
Thanks, is this last stage true by induction?