# Thread: Integral test for convergence

1. ## Integral test for convergence

I just took a math test, and there was one problem on it that I was having trouble with. (What makes it more ridiculous was that it was actually on the test twice written two different ways because of the nature of negative exponents...)

We had to test the convergence of the series

(from n=1 to infinity)
((n+1)^2)/(e^(n+1))

using the integral test.

It would be easy enough to do the ratio test, but technically we haven't learned that yet.

2. Originally Posted by rls
I just took a math test, and there was one problem on it that I was having trouble with. (What makes it more ridiculous was that it was actually on the test twice written two different ways because of the nature of negative exponents...)

We had to test the convergence of the series

(from n=1 to infinity)
((n+1)^2)/(e^(n+1))

using the integral test.

It would be easy enough to do the ratio test, but technically we haven't learned that yet.
The integral test requires that you investigate the convergence of:

$\displaystyle \int_{x=1}^{\infty} (x+1)^2e^{-(x+1)}\ dx$

To simplify this change the variable to $\displaystyle u=x+1$. Now the integrand may be written as the derivative of a quadratic in $\displaystyle u$ times $\displaystyle e^{-u}$ and so the corresponding indefinite integral can be integrated straight off to give a quadratic in $\displaystyle u$ times $\displaystyle e^{-u}$.

Which leaves a simple limit to evaluate to determine convergence of the integral and hence the sum.

CB