Evaluate the series, if it converges.
-$$9 + \frac { 18 } { 5 } + \frac { 36 } { 25 } + \frac { 72 } { 125 } + \ldots$$
A) 15
B) $\frac { 5 } { 3 }$
C) 6
D) Does not converge

Question

Quizplus · Accepted Answer

This series is a geometric series with the first term $a = 9$ and common ratio $r = \frac{18/5}{9} = \frac{2}{5}$. A geometric series converges if $|r| < 1$, which is true in this case. The sum of a converging geometric series is given by $S = \frac{a}{1 - r}$, so substituting $a = 9$ and $r = \frac{2}{5}$ gives $S = \frac{9}{1 - \frac{2}{5}} = \frac{9}{\frac{3}{5}} = 9 \times \frac{5}{3} = 15$.

Evaluate the Series, If It Converges 9+185+3625+72125+…9 + \frac { 18 } { 5 } + \frac { 36 } { 25 } + \frac { 72 } { 125 } + \ldots9+518​+2536​+12572​+…

Evaluate the Series, If It Converges $9 + \frac { 18 } { 5 } + \frac { 36 } { 25 } + \frac { 72 } { 125 } + \ldots$