Question

Find the sum of the convergent series 7 0.7 0.007

Answer 1

`7+0.7+0.007+\cdots=7\left(1+\frac1{10}+\frac1{100}+\cdots\right)`

Let
`S_n` denote the
`n`th partial sum of the series, i.e.


`S_n=1+\frac1{10}+\frac1{100}+\cdots+\frac1{100^n}`

Then


`\frac1{10}S_n=\frac1{10}+\frac1{100}+\frac1{1000}+\frac1{10^(n+1)}`

and subtracting from
`S_n` we get


`S_n-\frac1{10}S_n=\frac9{10}S_n=1-\frac1{10^(n+1)}`

`\implies S_n=\frac{10}9\left(1-\frac1{10^(n+1)}\right)`

As
`n\to\infty`, the exponential term vanishes, leaving us with


`\displaystyle\lim_(n\to\infty)S_n=\frac{10}9`

and so


`7+0.7+0.007+\cdots=7.777\ldots=\frac{70}9`