# I Summation of 1^1+2^2+3^3+...+k^k

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1. Jun 8, 2017

### ddddd28

Does that summatiom have a shorter representation at all?
$\sum_{n=1}^{k} n^n = ?$
I guess it is not of the form of constant power series, but I could not find an alternative.

Mentor note: made formula render properly

Last edited by a moderator: Jun 8, 2017
2. Jun 8, 2017

3. Jun 8, 2017

### Staff: Mentor

Huh? Looks both like nn to me.

I'm not aware of an analytic expression. It can probably be approximated with the Stirling formula and then some integration.

4. Jun 8, 2017

It's kk :-)

5. Jun 8, 2017

### Staff: Mentor

Maybe one can use Faulhaber to rewrite $n^n$ as difference of $\sum_{k=1}^n k^n - \sum_{k=1}^{n-1} k^n$ to get an expression in Bernoulli numbers which can then be summed again. A giant polynomial of Bernoulli numbers. Of course my bet to the original question
is NO. I mean the length of the expression is seven! Almost impossible to shorten.

6. Jun 8, 2017

### Staff: Mentor

Not true for an engineer for k>5 or so...

k^k

7. Jun 9, 2017

### Staff: Mentor

Now as you say it. Mathematicians can also shorter ...
$O(1)$