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

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 $a+b$.

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