Question

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Evaluate the series 1 + 0.1 + 0.01 + . . .

Answer 1

We can employ a simple repeated decimal trick:

`x=1.111\ldots`

`0.1x=0.111\ldots`

`\implies x-0.1x=1\implies0.9x=1\implies x=\frac1{0.9}=\frac{10}9`

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Alternatively, we can compute the partial sum of the series.

`\displaystyle S_n=\sum_(k=0)^n\frac1{10^k}`

`S_n=1+0.1+0.01+\cdots+\frac1{10^n}`

`0.1S_n=0.1+0.01+0.001+\cdots+\frac1{10^(n+1)}`

`\implies S_n-0.1S_n=0.9S_n=1-\frac1{10^(n+1)}`

`\implies S_n=\frac{10}9-\frac9{10^n}`

As
`n\to\infty`, the second term vanishes and we're left with
`\frac{10}9`. Notice that this is really just a more formal version of the earlier trick.

Answer 2

1.11

Explanation: