15. Power Series

Find the power series representation centered at $x=0$ of the following function. Give the interval of convergence for the resulting series. 

$f\left(x\right)=\frac{1}{1-x^3}$

Find the Taylor Series of $f\left(x\right)=\cos x$ centered $x=\pi$. Then, write the power series using summation notation.

Find the interval of convergence for the Taylor series for $f\left(x\right)=\sin x$ centered at $x=\frac{\pi}{2}$.

Find the interval of convergence for the Maclaurin series for $f\left(x\right)=\tan^{-1}x$

Find the Taylor polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\ln\left(x\right)$ centered at $x=1$.

Approximate $\ln1.5$ to four decimal places using the third-degree Taylor polynomial for $f\left(x\right)=\ln x$.

Find the Maclaurin polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\sin x$

Approximate $\sin0.3$ to four decimal places using the third-degree Maclaurin polynomial for $f\left(x\right)=\sin x$.