The sum of infinite geometric series is 25
Explanation:Given:
[tex]13)\text{ }Geometric\text{ series: }20,\text{ 4, 4/5, . . .}[/tex]To find:
the sum of the infinite series if it diverges
An infinite series converger if -1 < r < 1
We need to check if the value of r values in the above interval
[tex]\begin{gathered} common\text{ ratio = }\frac{next\text{ term}}{previous\text{ term}} \\ \\ common\text{ ratio = }\frac{4}{20}\text{ = 1/5} \\ \\ common\text{ ratio = }\frac{\frac{4}{5}}{4}\text{ = 1/5} \\ \\ \frac{1}{5}>\text{ -1 but less than 1 \lparen it falls in the interval\rparen} \\ Hence,\text{ it converges} \end{gathered}[/tex]The sum of geometric infinite series is given as:
[tex]\begin{gathered} S_{\infty}\text{ = }\frac{a}{1-r} \\ \\ a\text{ = 1st term = 20} \\ r\text{ = 1/5} \\ \\ S_{\infty}\text{ = }\frac{20}{1-\frac{1}{5}} \end{gathered}[/tex][tex]\begin{gathered} S_{\infty}\text{ = }\frac{20}{\frac{5-1}{5}}\text{ = }\frac{20}{\frac{4}{5}} \\ \\ S_{\infty}\text{ = 20 }\div\frac{4}{5} \\ \\ S_{\infty}\text{ = 20 }\times\frac{5}{4} \\ \\ S_{\infty}\text{ = 25} \end{gathered}[/tex]