Mathematics

Mathematics

Problem

The function hh is defined by h(x)=∑n=0∞(x+1)n+2(n+2)!h(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n+2}}{(n+2)!} for all real xx. Which option gives a power‐series expression for h′′(x)h''(x) centered at x=−1x=-1 along with the radius of convergence of h′′h''? a) h′′(x)=∑n=0∞(x+1)n(n+2)!h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n}}{(n+2)!}; radius of convergence ∞\infty b) h′′(x)=∑n=0∞(n+1)(x+1)nn!h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(n+1)(x+1)^{n}}{n!}; radius of convergence 11 c) h′′(x)=∑n=0∞(x+1)nn!h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n}}{n!}; radius of convergence ∞\infty d) h′′(x)=∑n=0∞(x+1)nn!h''(x)=\displaystyle \sum_{n=0}^{\infty}\frac{(x+1)^{n}}{n!}; radius of convergence 11

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