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
In the first part, we cannot really use the comparison theorem directly. In fact, the logarithm from 0 to 1/2 is negative, thus your inequality is in the wrong order. However, if it is multiplied by the minus, we arrive at the similar identity, which can now be used with the comparison theorem.