Simplest solution:
Spoiler:
Does converge?
I'm having trouble understanding my professor's solution:
exists because extends to a continuous function at x = 0.
Now this is where I'm having trouble. How are these equations derived?
Let be the floor function or greatest integer function. Then
.
Since for , the first term has a finite limit as , by the Alternating Series Test. The second term is bounded in absolute value by
,
so it converges to 0 by the squeeze theorem. So the limit exists, and therefore the integral converges.