Chapter 11.7, Problem 6MRE

To determine

To calculate:

The sum of each series

  $6+\frac{9}{2}+\frac{27}{8}+\mathrm{...}$

Answer to Problem 6MRE

The sum of each series is $S_{\infty }=24$

Explanation of Solution

Given information:

The given series is,

  $6+\frac{9}{2}+\frac{27}{8}+\mathrm{...}$

Formula:

The formula of sum of infinite series is,

  $S_{\infty }=\frac{a_1}{1-r}$

Where,

  $r:$ Common Ratio

  $S_{\infty }:$ Infinite term

  $a_1:$ First Term

Calculation:

The given series is,

  $6+\frac{9}{2}+\frac{27}{8}+\mathrm{...}$

Here in the above series by observing it can be observed that it has a common ratio. It means that the series is geometric series.

The common ratio of the above series = $\frac{3}{4}$ .

Here, the first term is = $6$

Now, by using the formula of sum for infinite geometric series.

  $\begin{array}{l}{S}_{\infty }=\frac{6}{\left(1-\frac{3}{4}\right)}\\ =\frac{6}{\left(\frac{4-3}{4}\right)}\\ =6\times \left(\frac{1}{4}\right)\\ =6\times 4=24\end{array}$

This is the sum of infinite geometric series.