Chapter 12.3, Problem 58E

To determine

Find whether the given infinite geometric series $1-1+1-1+\mathrm{....}$ is convergent or divergent if it is convergent then find its sum.

Answer to Problem 58E

The given infinite geometric series $1-1+1-1+\mathrm{....}$ is divergent

Explanation of Solution

Given information:

It is given in the question that, infinite geometric series $1-1+1-1+\mathrm{....}$

Formula used:

  $a_n=a{r}^{n-1}$

Where,

  $\begin{array}{l}a=firstterm\\ r=commonratio\\ a_{n}isthenthterm\\ n=numberofterm\end{array}$

Calculation:

  $1-1+1-1+\mathrm{....}$ means

  $\begin{array}{l}{a}_1=1\Rightarrow a=1\\ a_2=-1\Rightarrow ar=-1\Rightarrow r=\frac{-1}{1}\Rightarrow r=-1\end{array}$

  $a=1\&r=-1$

Here $\left|r\right|\ge 1$ so, this geometric series is divergent