30.8. a) Let $a_{n}=\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{2 n}$. Compute the limit $\lim _{n \rightarrow \infty} a_{n}$.

b) Let $b_{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots-\frac{1}{2 n}$. Prove that $\lim _{n \rightarrow \infty} b_{n}=\ln 2$.
30.8. a) Answer: $\ln 2$. According to problem 28.49 b) $\ln \frac{k+1}{k}<\frac{1}{k}<\ln \frac{k}{k-1}$. Adding such inequalities for $k=n, n+1, \ldots, 2 n$, we get $\ln \frac{2 n+1}{n}<a_{n}<\ln \frac{2 n}{n-1}$. Therefore, $\lim _{n \rightarrow \infty} a_{n}=\ln 2$.

Remark. Another proof can be found in the solution to problem 29.51.

b) According to problem $13.4 b_{n}=a_{n}$.