Problem: Show that the integral converges.
Solution: Consider the intervals for . We can rewrite the given integral as
where is some unimportant constant. So how can we go about bounding the integral
Well, first note that so we can say
Then, putting the last expression under a common denominator, we get
which we can easily bound with and . This gives us
Hence we know that
and this converges by a -series test. QED.
Comment: A pretty neat problem, though it is a standard convergence/divergence exercise. I’m sure there are many ways of doing this, but it’s always nice to come up with a cool way of showing that a series converges or diverges. It’s also interesting to note that the practice problem integral, which is only slightly different from this one, diverges.
Practice Problem: Show that the integral diverges.