Chapter 11.4, Problem 32E

Use Exercise 30 to show that the series ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{3}{k^2}}$ is convergent.Then use the comparison test to show that the series ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{e^{1/k}}{k^2}}$ is convergent.

Exercise 30:Let ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{a}_k}$ be a convergent series with sum $S$, and let $c$ be a constant. Then, ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}c{a}_k}$ is convergent series whose sum is $c\cdot S$. Make geometric picture to illustrate why this is true $c=2$ and the terms $a_k$ are all positive.