Chapter 5.3, Problem 185E

The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise.

185. In certain applications of probability, such as the so—called Watterson estimator for predicting mutation rates in population genetics, it is important to have an accurate

estimate of the number $H_k=\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{...}+\frac{1}{k}\right)$ Recall that $T_k=H_k-\ln k$ is decreasing. Compute $T=\underset{k\to \infty }{\lim }{T}_k$to four decimal places. (Hint:

$\frac{1}{k+1}<{\displaystyle {\int }_k^{k+1}\frac{1}{x}dx}$ .)