Chapter 7.1, Problem 77E

Harmonic sum In Chapter 10, we will encounter the harmonic sum $1\left|\frac{1}{2}\right|\frac{1}{3}|\cdots |\frac{1}{n}$. Use a left Riemann sum to approximate ${\displaystyle {\int }_1^{n+1}\frac{dx}{x}}$ (with unit spacing between the grid points) to show that $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}>\ln (n+1)$. Use this fact to conclude that $\underset{n\to \infty }{\lim }\left(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}\right)$ does not exist.