Thread: 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

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 denote the sum of the first n terms of the series. Then 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 converges to the same limit L as the Leibniz series.

Next, the (2n+1)th term in the series is , which converges to 0 as Therefore

Finally, since and converge to the same limit L, it follows that the whole series converges to L.

Originally Posted by Opalg

Finally, since and converge to the same limit L, it follows that the whole series converges to L.

Thanks, is this last stage true by induction?