Chapter 12.3, Problem 59AYU

To determine

To find: Whether the geometric series converges or diverges and find the sum if converges.

Answer to Problem 59AYU

The series is convergent and the sum of series is $\frac{20}{3}$ .

Explanation of Solution

Given information:

The series is ${\displaystyle \sum _{k=1}^{\infty }5{\left(\frac{1}{4}\right)}^{k-1}}$ .

Calculation:

For the given series $a_1=5$ , and $r=\frac{1}{4}$

Since $\left|r\right|<1$ the series is convergent.

The sum of infinite series which is convergent is given by,

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{5}{1-\frac{1}{4}}\\ =\frac{5}{\frac{3}{4}}\\ =\frac{20}{3}\end{array}$

Therefore, the series is convergent and the sum of series is $\frac{20}{3}$ .