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

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Answer 1 · 2017-06-12T23:32:06

The integral:

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

is convergent.

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.