Chapter 11.4, Problem 47PPS

To determine

To find the sum of the infinite series (if it exists):

  $\frac{15}{8}+\frac{5}{2}+\frac{10}{3}+\mathrm{......}$

Answer to Problem 47PPS

The sum does not exist.

Explanation of Solution

Given:

  $\frac{15}{8}+\frac{5}{2}+\frac{10}{3}+\mathrm{......}$

Concept Used:

1. Sum of an infinite series $a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}$ , whose first term is a and common ratio is r such that $\left|r\right|<1$is:

  ${\displaystyle {\sum }_{n=0}^{\infty }a\cdot r^n}=\frac{a}{1-r}$

2. Sum of an infinite series $a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}$ , whose first term is a and common ratio is $r>1$ does not exist.

Calculation:

In order to find the sum of infinite series

  $\frac{15}{8}+\frac{5}{2}+\frac{10}{3}+\mathrm{......}$

First find its first term and common ratio.

Note from the given infinite series that first term is $\frac{15}{8}$ , and the common ratio ( r ) can be find by dividing second term by first term, as

  $r=\frac{a_2}{a_1}$

Since, here

  $a_2=\frac{5}{2},a_1=\frac{15}{8}$

Thus,

  $\begin{array}{c}r=\frac{\frac{5}{2}}{\frac{15}{8}}\\ =\frac{5}{2}\times \frac{8}{15}\\ =\frac{4}{3}\\ =1.33\end{array}$

Since, the common ratio of the series is greater than 1.

Thus, the sum of the infinite series does not exist.