Solutions - 0230-02

$$\begin{array}{c}f(x)=\sum _{n=0}^{\infty }{(-1)}^n\frac{x^{2n+2}}{n+1}\\ \\ \gg \gg \frac{d}{dx}f(x)=\frac{d}{dx}\sum _{n=0}^{\infty }{(-1)}^n\frac{x^{2n+2}}{n+1}\\ \\ \gg \gg \sum _{n=0}^{\infty }{(-1)}^n\frac{2n+2}{n+1}{x}^{2n+1}\gg \gg \sum _{n=0}^{\infty }{(-1)}^n(2)x^{2n+1}\end{array}$$

If $R=1$ for $f(x)$, then we know $R=1$ for $f^{\prime }(x)$.