Calculating $ \sum_{ k=1}^n (-1)^k k $

Question

I've got two sums here

$ \sum_{k=1}^n (-1)^k k $ and $ \sum_{k=1}^n (-1)^k k^2 $

I need to solve them through term isolation. I struggle to find anything similar. So I am asking you for help; do you have any ideas or approaches?

A similar idea — grouping pairs of terms as far as possible — works for the second sum too. — Steven Stadnicki, May 17 '21 at 06:09
I did consider that as well. Though the "term isolation" confuses me. Does that just mean consindering each consecutive pair? — wondering1123, May 17 '21 at 06:10

score 1 · Answer 1 · answered May 17 '21 at 06:06

Try writing out some terms as $n$ grows.

$$\left\{\sum_{k=1}^n(-1)^kk\right\}_{n=1}^{\infty}=\{-1,1,-2,2,-3,3\ldots\}$$

You have the ceiling of $n/2$ alternating negative and positive.

$$\left\{\sum_{k=1}^n(-1)^kk^2\right\}_{n=1}^{\infty}=\{-1,3,-6,10,-15,21\ldots\}$$

You have the triangular numbers $n(n+1)/2$ alternating negative and positive.

If you need a proof, there are good ideas in the comments.

1 Answers1