$(10\mathrm{.}08HC)$

The function $h$ is defined by the power series $h(x)=x+\frac{x^2}{3}+\frac{x^3}{5}+\frac{x^4}{7}+\frac{x^5}{9}+$dots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{x^{n+1}}{2n+1}$

Part A: Determine the interval of convergence of the power series for $h.(10$ points$)$

Part B: Find $h^{\text{'}}(-1)$ and determine if the new series converges or diverges. Justify your response and discuss the relationship between the radius of convergence and interval of convergence of $h(x)$ and $h^{\text{'}}(x).(10$ points$)$

Part C: Determine if the power series $h(x^2)$ has any points of inflection. Justify your answer. $(10$ points$)$