Is there any closed form for $\sum_{k=1}^n \frac{1}{k^k}$?

Question

Is there any closed form for the summation: $$\sum_{k=1}^n \frac{1}{k^k} = ? $$ or at least a tight lower bound?

Answer 1

J. Bernoulli showed that

$$\sum_{n=1}^\infty n^{-n}= \int_0^1 x^{-x}\,dx$$

This result is often called the "Sophomore's dream" because it looks too good to be true.

I don't believe there is a form more "closed" than this.