Check that $a_n = \sum_{k=1}^n \frac{1}{n+k}$ is bounded from above by $\frac 34$ Any ideas on how to check that
$
a_n = \sum_{k=1}^n\frac{1}{n+k} \le \frac 34\,?
$
It's pretty easy to see that the sequence is monotonic and has an almost trivial upper bound of $1$ (and a trivial lower bound of $1/2$).
I solved it several years ago but can't recall my solution. I do remember I scratched my head for a while, though. I'm quite sure one needs to get an upper bound through some algebraic manipulation to get $\sum_{k=2}^\infty \frac {1}{k^2}$ (well, some polynomial of order 2 in $k$) but can't recover the proper way to do it (without relying on integral calculus).
Wolframalpha suggests that $\lim_n a_n = \ln(2)$, which can be easily seen by integration. Is there a `more' elementary way to see it? My guess would be no.

you can use this theorem, for the case a = 1, to prove the inequality.

