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-\frac53+\frac59-\frac5{27}+\cdots+5\left(-\frac13\right)^(n-1)+5\left(-\frac13\right)^n`

Multiplying both sides by
`-\frac13`, we have


`-\frac13S_n=-\frac53+\frac59-\frac5{27}+\cdots+5\left(-\frac13\right)^n+5\left(-\frac13\right)^(n+1)`

Subtracting the second equation from the first gives


`S_n-\left(-\frac13\right)S_n=5-5\left(-\frac13\right)^(n+1)`

`\frac43S_n=5-5\left(-\frac13\right)^(n+1)`

`S_n=\frac{15}4\left(1-\left(-\frac13\right)^(n+1)\right)`

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


`S_\infty=\frac{15}4`