A probability-generating function is modeled by the series S(x)= _ n=0 ^ ( x-1 4 )^ n

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

A probability-generating function is modeled by the series S(x)=∑n=0∞(x−14)n,S(x)=\sum_{n=0}^{\infty}\left(\frac{x-1}{4}\right)^{n}, where xx is a real variable. (a) Find a closed-form expression for S(x)S(x). (b) Determine the radius and complete interval of convergence for SS. (c) Evaluate the definite integral ∫12S(x) dx\displaystyle \int_{1}^{2} S(x)\,dx, or explain why the integral does not exist.

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