(Solved): (a) Use the definition of Taylor series centered at zero, shown below, to derive the Taylor series ...

(a) Use the definition of Taylor series centered at zero, shown below, to derive the Taylor series for $f(x)=1/(1+x)$. Write your result in sigma notation. $P_{?}(x)=f(0)+\frac{f^{?}(0)}{1!}x+\frac{f^{??}(0)}{2!}{x}^2+\frac{f^{???}(0)}{3!}{x}^3+?+\frac{f^{(n)}(0)}{n!}{x}^n+?$ (b) Replace $x$ by $x^2$ in the result (a), to derive a Taylor series for $1/\left(1+x^2\right)$. (c) Integrate the series you obtained in (b), to derive a Taylor series for $\arctan x$. Tip: Integration will produce a constant of integration, $C$. After you integrate, find the value of $C$.