Chapter 8, Problem 14CR

To determine

To find: the sigma notation of the series and sum of the series.

Answer to Problem 14CR

Sigma notation of the series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}\mathrm{..........}+\frac{9}{10}$ is $\underset{\_}{{\displaystyle \sum _{k=1}^9\frac{k}{k+1}}}$ and sum of the series is approximately $\underset{\_}{7.07}$ .

Explanation of Solution

Given information:

The series which needs to be represent as sigma notation is as follows

  $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}\mathrm{..........}+\frac{9}{10}$

Calculation:

The series which needs to be represent as sigma notation is as follows

  $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}\mathrm{..........}+\frac{9}{10}$

  $=\frac{1}{1+1}+\frac{2}{2+1}+\frac{3}{3+1}\mathrm{..........}+\frac{9}{9+1}$

  $={\displaystyle \sum _{k=1}^9\frac{k}{k+1}}$

Calculating sum using utility

  

Hence sigma notation of the series $\frac{1}{2}+\frac{2}{3}+\frac{3}{4}\mathrm{..........}+\frac{9}{10}$ is $\underset{\_}{{\displaystyle \sum _{k=1}^9\frac{k}{k+1}}}$ and sum of the series is $\underset{\_}{7.07}$ .