This entry devotes a similar question that raises during a course of series. We all know that for a convergent series of (positive) real number
it is necessary to have
This is the so-called -th term test. A natural extension is the following question
Question. Suppose is positive on and
exists. Must tend to zero as ?
I will show that this is indeed not the case. Take
The above solution tell us the fact that the convergence of improper integral is not enough to guarantee .
It turns out to propose the question under what condition on , tend to zero as ? There are several answers in the literature, for example
- If is positive, differentiable and is uniformly bounded, say by , then by the mean value theorem, we can prove .
- If is uniformly continuous on then the same conclusion still holds by the Cauchy theorem.
- If is monotone decreasing, we obtain a stronger result, i.e. .