# Descendant Series

Calculus Level pending

$\large Z(x) = \sum_{n=1}^\infty n(n-1)(n-2) x^{n-3}$

Let $$Z(x)$$ be a function as described above for $$|x| < 1$$. If the prime factorization of $$Z\left(\dfrac12\right)$$ is $$a^c \times b$$, where $$a,b$$ and $$c$$ are prime numbers, find $$\dfrac{a+b}c$$.

Give your answer to 1 decimal place.

Hint: The following convergent geometric series may prove useful: $$\displaystyle \sum_{n=1}^\infty x^n= \dfrac1{1-x}$$.

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