Question

Find the sum of the infinite geometric series: 5-(5)/(3)+(5)/(9)-(5)/(27)...

Answer 1

Denote the $n$th partial sum of the series by

$S_n=5-\dfrac53+\dfrac59-\dfrac5{27}+\cdots+5\left(-\dfrac13\right)^{n-1}+5\left(-\dfrac13\right)^n$

Multiplying both sides by $-\dfrac13$, we have

$-\dfrac13S_n=-\dfrac53+\dfrac59-\dfrac5{27}+\cdots+5\left(-\dfrac13\right)^n+5\left(-\dfrac13\right)^{n+1}$

Subtracting the second equation from the first gives

$S_n-\left(-\dfrac13\right)S_n=5-5\left(-\dfrac13\right)^{n+1}$
$\dfrac43S_n=5-5\left(-\dfrac13\right)^{n+1}$
$S_n=\dfrac{15}4\left(1-\left(-\dfrac13\right)^{n+1}\right)$

As $n\to\infty$, the geometric term approaches 0, leaving you with

$S_\infty=\dfrac{15}4$