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.

Read more about this topic:  142857 (number)

Famous quotes containing the words infinite and/or sum:

    “Moreover, the universe as a whole is infinite, for whatever is limited has an outermost edge to limit it, and such an edge is defined by something beyond. Since the universe has no edge, it has no limit; and since it lacks a limit, it is infinite and unbounded. Moreover, the universe is infinite both in the number of its atoms and in the extent of its void.”
    —Epicurus (c. 341–271 B.C.)

    “No, the five hundred was the sum they named
    To pay the doctor’s bill and tide me over.
    It’s that or fight, and I don’t want to fight
    I just want to get settled in my life....”
    —Robert Frost (1874–1963)