Is it appropriate for MO? The question is rather elementary and originated from elementary calculus/analysis. As I've said, it's not hard to determine the asymptotic behavior if we look at the closed form, but I want to enrich my means of manipulating summations. It seems a good example for me to estimate a summation whose tail is small but not that small.

If it's appropriate, I'll migrate it here as soon as the bounty is expired (and of course, if there's no answer in MSE).