Chapter 12.3, Problem 59E

To determine

Find whether the given infinite geometric series $3-3\left(1.1\right)+3{\left(1.1\right)}^2-3{\left(1.1\right)}^3+\mathrm{....}$ is convergent or divergent if it is convergent then find its sum.

Answer to Problem 59E

The given infinite geometric series $3-3\left(1.1\right)+3{\left(1.1\right)}^2-3{\left(1.1\right)}^3+\mathrm{....}$ is divergent

Explanation of Solution

Given information:

It is given in the question that, infinite geometric series $3-3\left(1.1\right)+3{\left(1.1\right)}^2-3{\left(1.1\right)}^3+\mathrm{....}$

Formula used:

  $a_n=a{r}^{n-1}$

Where,

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

Calculation:

  $3-3\left(1.1\right)+3{\left(1.1\right)}^2-3{\left(1.1\right)}^3+\mathrm{....}$ means

  $\begin{array}{l}{a}_1=3\Rightarrow a=3\\ a_2=3\left(1.1\right)\Rightarrow ar=3\left(1.1\right)\Rightarrow r=\frac{3\left(1.1\right)}{3}\Rightarrow r=\left(1.1\right)\end{array}$

  $a=3\&r=1.1$

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