User:Thegeebe2/Sandbox

10100/970299 is a rational number that expresses the decimal .010409162536496482... in exact form.

The decimal is a pattern created by placing the squares of increasing integers every two decimal places. It is generated by the series

$$ \sum_{n=1}^\infty (.01)^n(n^2)$$

Derivation
One type of derivation uses a method similar to that which is used to approximate repeating decimals by converting the decimal into repeating decimal form.

\begin{align} x & = .0104091625364964\dots \\ 100x  & = 1.040916253649\dots\\ 100x-x = 99x & = 1.03050709111315\dots\\ 9900x & = 103.050709111315\dots\\ -99x & = -1.03050709111315\dots\\ 9801x & = 102.0202020202\dots\\ 980100x & = 10202.020202020\dots\\ -9801x & = -102.0202020202\dots\\ 970299x & = 10202.\overline{02} - 102.\overline{02} = 10202-102 = 10100\\

x & = \frac{10100} {970299}

\end{align}