Does this sum have a closed formula?

Question

$$c^1 + c^2 + \cdots + c^n = c\frac{c^n - 1}{c-1}$$

Right?

What about $c^\frac{1}{m} + c^\frac{2}{m}+ ... c^\frac{n}{m}$? Does it also have a closed formula?

Thanks a lot!

Answer 1

Let $d = c^{\frac{1}{m}}$. Then your sum is $$ d + d^2 + \ldots + d^m, $$ which you already know how to sum up. Then substitute back in for $d$.