Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

a. Determine whether the series $\displaystyle\sum_{n=1}^\infty (-1)^n\ln\left(1+\frac{1}{n}\right)$ diverges or converges conditionally or converges absolutely and give reasons for your answer. (6 points)

b. Show that if $\displaystyle\sum_{n=1}^\infty a_n$ converges, then $\displaystyle\sum_{n=1}^\infty \left(\frac{3+\sin(a_n)}{5}\right)^n$ converges. (6 points)