Chapter 5.6, Problem 360E

The following series converge by the ratio test. Use

summation by parts,

   ${\displaystyle \underset{k=1}{\overset{n}{\sum }}{a}_k\left(b_{k+1}-b_k\right)=\left[a_{n+1}{b}_{n+1}-a_1{b}_1\right]-{\displaystyle \underset{k=1}{\overset{n}{\sum }}{b}_{k+1}\left(a_{k+1}-a_k\right)}}$

to find the sum of the given series.

360. ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{k}{2^k}}$ (Hint: Take $a_k=k$ and $b_k=2^{1-k}$ .)