Question

What is the sum of the infinite geometric series 9−6+4−83+...9-6+4-83+...??

Answer 1

Successive terms have a common factor of
`-\frac23`, which means the
`n`th partial sum of the series can be written as


`S_n=9-6+4-\frac83+\cdots+9\left(-\frac23\right)^(n-1)`

`S_n=9\left(1+\left(-\frac23\right)^1+\left(-\frac23\right)^2+\left(-\frac23\right)^3+\cdots+\left(-\frac23\right)^(n-1)`


`\implies-\frac23S_n=9\left(\left(-\frac23\right)^1+\left(-\frac23\right)^2+\left(-\frac23\right)^3+\left(-\frac23\right)^4+\cdots+\left(-\frac23\right)^n\right)`


`\implies S_n-\left(-\frac23\right)S_n=9\left(1-\left(-\frac23\right)^n\right)`

`\frac53S_n=9\left(1-\left(-\frac23\right)^n\right)`

`S_n=\frac{27}5\left(1-\left(-\frac23\right)^n\right)`

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


`S=\displaystyle\lim_(n\to\infty)S_n=\frac{27}5`