Question $1$

Let $f(x)=\frac{x}{4x-2x^2-1}.$ Find the Taylor series for $f$ centered at $x=1.$

Question $2$

Suppose we use a $3^{rd}-$degree Taylor polynomial, $T_3(x),$ centered at $x=9$ to approximate $\sqrt[\phantom{2}]{10}.$ What is the maximum error in the approximation?

Question $3$

Previously we found that $cos(x)={\displaystyle \underset{x=0}{\overset{}{}}}\frac{(-1{)}^n{x}^{2n}}{(2n)!}.$ Prove that the Maclaurin series actually converges to $cos(x).($Be sure to show your own work.$)$