Question

In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.

Short answer

• The common ratio $r$lies between $-1<r<1$then we can have the sum of an infinite geometric series.
• The sum of an infinite geometric series is$S=\frac{a}{1-r}$

Step-by-step solution

Step 1. Given information

To determine when the infinite geometric series have the sum and how to find it.

Step 2. To determine when infinite geometric series have sum

The infinite geometric series is

$a+ar+a{r}^2+a{r}^3+·············$

where

$$\begin{gathered} a=firsttermofthesequence \\ r=commonratio \end{gathered}$$

The common ratio $r$lies between $-1<r<1$then we can have the sum of an infinite geometric series.

Step 3. To determine the sum of infinite geometric series 

The sum $S$of an infinite geometric series is given by

$S=\frac{a}{1-r}$

where

$$\begin{gathered} a=firsttermofthesequence \\ r-commonratio \end{gathered}$$