Does $\int_1^\infty x^{-x}$ converges?

Question

I want to use the comparison test - I thought of $1 \over x$. but this function is larger and also diverges so it doesn't help me much.

Thanks in advance!

Answer 1

If $x>2$ hen $$ \frac{1}{x^x}<\frac{1}{x^2}. $$