Formula for the following sum?

Question

I simply wonder if it exists a formula for the sum $S_n = \sum_{k=1}^{n} k^k$ ? If it does, then what is it? If not, how do we know that?

Answer 1

According to wolfram, there is no known closed form formula for the anti-derivative of $x^x$, so probably none exists. (There is an algorithm to check if a function has an anti-derivative in terms of elementary functions, which can confirm that $x^x$ has no simple anti-derivative.) In general, formulas for integrals are typically easier to obtain than formulas for the corresponding series. So likely no easy expression for your sum will be found. However, I'm not aware of any algorithms to determine whether a series has a formula like can be done for anti-derivatives.