Proofs of $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$

Question

Prove that $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$. This comes from this question Evaluate $\sum_0^\infty \frac{1}{n^n}$.

I agree. It's a duplicate. – user103828 Mar 08 '14 at 15:53 — user103828, Mar 08 '14 at 15:53

score 1 · Accepted Answer · edited Apr 13 '17 at 12:21

1

This identity is called Sophomore's dream (a "dream" because it seems too good to be true, and at the same time an identity because it is indeed true!).

It's proof can be found in Pedro Tamaroff's answer to this related (actually, its exactly the same) question.

edited Apr 13 '17 at 12:21

Community

1

answered Mar 08 '14 at 15:44

Yiyuan Lee

14,435

Proofs of $\sum_{n=1}^\infty \frac{1}{n^n}=\int_0^1 \frac{1}{x^x}dx$

1 Answers1