Chapter 10.4, Problem 60HP

To determine

To explain who is correct in calculating the sum of infinite series

  $1-1+1-\mathrm{.....}$

Answer to Problem 60HP

Austin is right.

Explanation of Solution

Given:

  

Concept Used:

1. Sum of an infinite geometric 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

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

2. Sum of an infinite geometric 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 $r>1$is divergent(doesn’t exist).

Calculation:

In order to find the sum of infinite series

  $1-1+1-\mathrm{.....}$

First check whether this sum exists or not, for that check if the common ratio of the series satisfies $\left|r\right|<1$ or $r>1$ .

Now, the common ration of the series( r ) can be find by dividing the second term by the first term, as

  $r=\frac{a_2}{a_1}$

Since, here

  $a_2=-1,a_1=1$

Thus,

  $\begin{array}{c}r=\frac{-1}{1}\\ =-1\end{array}$

Since,

  $\begin{array}{c}\left|r\right|=\left|-1\right|\\ =1\\ \ge 1\end{array}$

Since the common ratio, r , is equals to 1.

Thus, the given infinite geometric series is divergent, thus Austin is right.