Test the following improper integral for convergence:
The integral converges.
Proof. We recall the definition of the hyperbolic cosine in terms of the exponential,
Then we have
The middle integral converges since it is a proper integral of an integrable function. For the integral on the left we have
But, we have
for all . We know converges by Example 4 (page 418 of Apostol); hence,
converges. Since this is also the third integral in the sum above, we have proved that every integral in the sum converges, and hence we have established the convergence of