Question

How do you use the integral test to determine whether `int x^-x` converges or diverges from `[1,oo)`?

Answer 1

The integral:

`int_1^oo x^(-x)dx`

is convergent.

Explanation:

Based on the integral test, the convergence of the integral:

`int_1^oo x^(-x)dx`

is equivalent to the convergence of the series:

`sum_(n=1)^oo n^(-n)`

the series has positive terms and we can apply the root test:

`L = lim_(n->oo) root(n)(n^(-n)) = lim_(n->oo) (n^(-n))^(1/n) = lim_(n->oo) n^-1 = lim_(n->oo) 1/n = 0`

As `L < 1` the series is convergent and than also the integral is convergent.