Test the following improper integral for convergence:

The integral converges.

*Proof.* First, we write

For the first integral we know for all we have ; hence,

Then, the integral

(We know by L’Hopital’s, writing or by Example 2 on page 302 of Apostol.) Hence,

converges by the comparison theorem (Theorem 10.24 on page 418 of Apostol).

For the second integral, we use the expansion of about ,

Then we have

But this integral converges since it has no singularities.

Thus, we have established the convergence of