Chapter 8.1, Problem 113E

To determine

To evaluate: the fourth partial sum of the given infinite series.

Answer to Problem 113E

Fourth partial sum of ${\displaystyle \sum _{i=1}^{\infty }\frac{7}{5^i}}$ is ${\displaystyle \sum _{i=1}^4\frac{7}{5^i}}=\frac{1086}{625}$ .

Explanation of Solution

Given information:

A summation expression is given as

  ${\displaystyle \sum _{i=1}^{\infty }\frac{7}{5^i}}$

Formula used:

The nth partial sum of ${\displaystyle \sum _{i=1}^{\infty }{a}_k}$ is ${\displaystyle \sum _{i=1}^n{a}_k}$ .

Calculation:

Consider the given sum.

  ${\displaystyle \sum _{i=1}^{\infty }\frac{7}{5^i}}$

Now, the fourth partial sum of ${\displaystyle \sum _{i=1}^{\infty }\frac{7}{5^i}}$ will be ${\displaystyle \sum _{i=1}^4\frac{7}{5^i}}$ .

Now, substitute 1, 2, 3, and 4 for $i$ in the expression $\frac{7}{5^i}$ and add those.

  $\begin{array}{c}{\displaystyle \sum _{i=1}^4\frac{7}{5^i}}=\frac{7}{5^1}+\frac{7}{5^2}+\frac{7}{5^3}+\frac{7}{5^4}\\ =\frac{7}{5}+\frac{7}{25}+\frac{7}{125}+\frac{7}{625}\\ =\frac{875+175+35+1}{625}\\ =\frac{1086}{625}\end{array}$

Hence, the fourth partial sum of ${\displaystyle \sum _{i=1}^{\infty }\frac{7}{5^i}}$ is ${\displaystyle \sum _{i=1}^4\frac{7}{5^i}}=\frac{1086}{625}$ .