142857 (number) - 1/7 As An Infinite Sum

1/7 As An Infinite Sum

There is an interesting pattern of doubling, shifting and addition that gives 1/7.

\begin{align}1/7\ & = 0.142857142857142857\ldots \\ & = 0.14 + 0.0028 + 0.000056 + 0.00000112 + 0.0000000224 + 0.000000000448 + 0.00000000000896 + \cdots \\ & = \frac{14}{100} + \frac{28}{100^2} + \frac{56}{100^3} + \frac{112}{100^4} + \frac{224}{100^5} + \cdots + \frac{7\times2^N}{100^N} \\ & \left(= \frac{7}{50} + \frac{7}{50^2} + \frac{7}{50^3} + \frac{7}{50^4} + \frac{7}{50^5} + \cdots + \frac{7}{50^N} \right) \end{align}

Each term is double the prior term shifted two places to the right.

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