logarithm function

Quote:

Originally Posted by elmo

Let n be a positive integer greater than 2.
a)Find the greatest integer k for which
1/2 + 1/3 + ... + 1/k < ln(n)

b)Find the least integer k for which
ln(n) < 1 + 1/2 + 1/3 + ... + 1/k

This is Napier’s inequality: $\displaystyle 0 < a < b\; \Rightarrow \;\frac{1}{b} \leqslant \frac{{\ln (b) - \ln (a)}}{{b - a}} \leqslant \frac{1}{a}$.
From which it follows that: $\displaystyle \frac{1}{{N + 1}} \leqslant \ln (N + 1) - \ln (N) \leqslant \frac{1}{N}$.

So we get $\displaystyle \sum\limits_{k = 1}^{N - 1} {\frac{1}{{k + 1}}} \leqslant \ln (N) \leqslant \sum\limits_{k = 1}^{N - 1} {\frac{1}{k}} $.