15. Power Series

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

Approximate $\(\sin\)0.3$ to four decimal places using the third-degree Maclaurin polynomial for $f\(\left\)(x\(\right\))=\(\sin\) x$.

Approximate $\(\ln\)1.5$ to four decimal places using the third-degree Taylor polynomial for $f\(\left\)(x\(\right\))=\(\ln\) x$.

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

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

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