Chapter 12.3, Problem 62E

To determine

Find whether the given infinite geometric series $1-\sqrt{2}+2-2\sqrt{2}+4-\mathrm{....}$ is convergent or divergent if it is convergent then find its sum.

Answer to Problem 62E

The given infinite geometric series $1-\sqrt{2}+2-2\sqrt{2}+4-\mathrm{....}$ is divergent

Explanation of Solution

Given information:

It is given in the question that, infinite geometric series $1-\sqrt{2}+2-2\sqrt{2}+4-\mathrm{....}$

Formula used:

  $a_n=a{r}^{n-1}$

Where,

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

Calculation:

  $1-\sqrt{2}+2-2\sqrt{2}+4-\mathrm{....}$ means

  $\begin{array}{l}{a}_1=1\Rightarrow a=1\\ a_2=-\sqrt{2}\Rightarrow ar=-\sqrt{2}\Rightarrow r=\frac{-\sqrt{2}}{1}\Rightarrow r=-\sqrt{2}\end{array}$

  $a=1\&r=-\sqrt{2}$

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