What is the value of $\displaystyle \sum _{n=1}^{\infty} \frac{n^{10}}{2^{n}}$ ?

The answer is 204495126.

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This can be easily done using poylogarithms,

$polylog[-10,2] = -204495126$

But since the sum can never be negative,

Hence, $\boxed{204495126}$