Chapter 11.4, Problem 5CYU

To determine

To determine whether the below infinite geometric series is convergent or divergent

  $440+220+110+\mathrm{....}$

Answer to Problem 5CYU

Convergent

Explanation of Solution

Given:

  $440+220+110+\mathrm{....}$

Concept Used:

Sum of an infinite geometric progression $a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}$ , whose first term is a and common ratio is r is:

1. Convergent if $\left|r\right|<1$
2. Divergent if $1\le \left|r\right|$

Calculation:

In order to determine whether the below infinite geometric series is convergent or divergent

  $440+220+110+\mathrm{....}$

Check whether the modulus of common ratio of the series $\left|r\right|<1$ or $1<\left|r\right|$ .

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

  $a_2=220,a_1=440$

Thus,

  $\begin{array}{c}r=\frac{220}{440}\\ =\frac{1}{2}\\ =0.5\end{array}$

Since,

  $\begin{array}{c}\left|r\right|=\left|0.5\right|\\ =0.5\\ <1\end{array}$

Since the common ratio, r , is less than 1.

Thus, the given infinite geometric series is convergent.