It Doesn't Telescope

Algebra Level 3

\[S= \frac{1^{2}}{10}+\frac{2 \times 1^{2}+2^{2}}{10^{2}}+\frac{3 \times 1^{2}+2 \times 2^{2}+3^{2}}{10^{3}}+\cdots \]

The sum \(S\) defined above is an infinite sum whose \(n^\text{th}\) term is \[\dfrac{n \times 1^{2}+(n-1) \times 2^{2}+(n-2) \times 3^{2} + \cdots + n^{2}}{10^{n}}\] for \(n=1,2,\ldots \).

If \(S\) can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


Inspiration.

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