What is the formula for the sum: #1+22+333+4444+...+overbrace(nnn...n)^"n terms"# ?

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Answer 1 · 2017-09-06T21:07:47

#sum_(k=1)^n k^k#

There is no nice algebraic closed formula for this sum.

About the best we can do is provide some notation...

#1+2 * 2+3 * 3 * 3+4 * 4 * 4 * 4+...+overbrace(n * n * n * ... * n)^"n times" = sum_(k=1)^n k^k#

and note that:

#lim_(n->oo) (sum_(k=1)^n k^k)/n^n = 1#

What is the formula for the sum: #1+2*2+3*3*3+4*4*4*4+...+overbrace(n*n*n*...*n)^"n terms"# ?