For a fixed + integer k, what is the general formula for the finite series SUM j=1 to n of [(-1)^j][ j^k] ?
This is not an easy question.
Without the (-1)^j you can find the answer Summation - Wikipedia, the free encyclopedia
To figure out the answer you will probably need to use the same techniques.
Other than this I can only give you the hint that
$\displaystyle \sum _{j=1} ^n j^k$ is a polynomial of degree k+1, this means that you can find the closed formula by evaluating at k+2 points, and interpolating them (since there is a unique interpolating polynomial.)