Descendant Series

Calculus Level 3

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

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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