Q:

How do I solve this kind of sigma notation problem? 

How do I solve the following? ( by hand, its easy to find an answer with a calculator but I need an answer than can be done with some kind of rule/formula/identity). 
$$\sum_{n=1}^5 n^n$$
and is there a certain rule i can use (by hand of course) that will let me solve this with any value instead of 5? 
Thanks! 

A:

I do not think there is a nicer closed form for this; you basically have to sum this by hand or with a computer to find the answer.  Note that, for instance, it cannot be equal to any polynomial, because the terms grow faster than $x^k$ for any fixed $k$. Check out its entry in the Online Encyclopedia of Integer Sequences for more references.