Sigma with n=1 to n= positive infinity

Question

(x^3)* (e^(-x^4))

Does it converge or diverge and use the integral test to show how. I got that it diverges, but this isn't correct. Can someone please explain? Thanks.

Answer 1

I assume the expression is
(n^3)* (e^(-n^4)) for n=1 to +&inf;
You can use the ratio test to show that it converges, namely, show that
a_n+1/a_n < 1 ∀ k≥1.

The integral test will result in a definite integral from 1 to +&inf; which is not an elementary function, but evaluates numerically to about 0.09. However, this still shows that the series is convergent.

Answer 2

Integral from 1 to infinity of
(x^3)* (e^(-x^4)) dx =

1/4 exp(-1)