Question

Let \(S_{N}:=1+\frac{1}{2}+\cdots+\frac{1}{N}\). Find an estimate for \(N\) so that \(S_{N}>100\).

Answer 1

By the idea behind the integral test
\[
\ln (N+1)<S_{N}<1+\ln N
\]
Thus, to insure that \(S_{N}>100\) pick \(\ln (N+1)>100\), that is, \(N+1>\) \(e^{100} \approx 2.7 * 10^{43}\).
On the other hand, if \(1+\ln N<100\), then \(S_{N}<100\). Here we can pick any \(N<e^{99} \approx 10^{43}\).