Closed form for the sum 1^1+2^2+3^3+...+k^k

[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[/size]

 

[mathman]

Your question is for $$n^n$$, but the title line is for $$n^2$$. Which is it?

https://en.wikipedia.org/wiki/Faulhaber's_formula

 

[mfb]

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.

 

[cosmik debris]

It's k^k :-)

 

[fresh_42]

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

Does that summatiom have a shorter representation at all?

is NO. I mean the length of the expression is seven! Almost impossible to shorten.

 

[berkeman]

is NO. I mean the length of the expression is seven! Almost impossible to shorten.

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

Spoiler

k^k

 

[fresh_42]

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

Spoiler

k^k

Now as you say it. Mathematicians can also shorter ...

Spoiler

$O(1)$