AP Calculus BC 10.14 Finding Taylor or Maclaurin Series for a Function Questions

A function f has derivatives of all orders for -1<x<1. The derivatives of f satisfy the conditions above. The Maclaurin series for f converges to f(x) for |x|<1.

Show that the first four nonzero terms of the Maclaurin series for f are $x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}$, and write the general term of the Maclaurin series for f.

The Taylor series for a function f about x=4 is given by $\sum_{n=1}^{\infty} \frac{(x-4)^{n+1}}{(n+1) 3^{n}}=\frac{(x-4)^{2}}{2 \cdot 3}+\frac{(x-4)^{3}}{3 \cdot 3^{2}}+\frac{(x-4)^{4}}{4 \cdot 3^{3}}+\cdots+\frac{(x-4)^{n+1}}{(n+1) 3^{n}}+\cdots$ and converges to f(x) on its interval of convergence.

The Taylor series for $f^{\prime}$ described in part B is a geometric series. For all x in the interval of convergence of the Taylor series for $f^{\prime}$, show that $f^{\prime}(x)=\frac{x-4}{7-x}$.