I'm working on Project Euler, problem 48:
The series, 11 + 22 + 33 + ... + 1010 = 10405071317.
Find the last ten digits of the series, 11 + 22 + 33 + ... + 10001000.
This would be the equivalent of: $$ f(n) = \sum_{i = 1} ^ n i^i $$
The original question aside, it got me thinking. The rate of change from n to n+1 would be i^i, so if I could take the integral of ii, I could get a generalized formula for any given n. Upon plugging it into Wolfram Alpha, I got a monstrous expansion, but it said, "no result found in terms of standard mathematical functions". It's been years since I've taken calculus and I'm mostly a computer science guy to begin with, but is this premise sound? And if not, where did I go wrong?
There is no closed form for this integral, but there might be special integrals solving this, which can be only calculated numerically.