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