Search for summation formula

Question

Is there any closed formula for the sum $ ~\sum_{k = 1} ^ {n} r^k k^r ~$ where $~r~$ is an integer?

Thank you very much in advance.

Answer 1

The sum corresponds to the following polylogarithm:

$$ \sum_{k = 1} ^ {n} r^k k^r = \operatorname{Li}_{-r}(r)$$