Problem 86
In the following exercises, find the radius of convergence and the interval of convergence for the given series. $$ \sum_{n=0}^{\infty} \frac{3 n x^{n}}{12^{n}} $$
Problem 87
In the following exercises, find the radius of convergence and the interval of convergence for the given series. $$ \sum_{n=0}^{\infty} \frac{2^{n}}{e^{n}}(x-e)^{n} $$
Problem 88
In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series. $$ f(x)=\frac{x^{2}}{x+3} $$
Problem 89
In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series. $$ f(x)=\frac{8 x+2}{2 x^{2}-3 x+1} $$
Problem 90
In the following exercises, find the power series for the given function using term-by-term differentiation or integration. $$ f(x)=\tan ^{-1}(2 x) $$
Problem 91
In the following exercises, find the power series for the given function using term-by-term differentiation or integration. $$ f(x)=\frac{x}{\left(2+x^{2}\right)^{2}} $$
Problem 92
In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation? $$ f(x)=x^{3}-2 x^{2}+4, a=-3 $$
Problem 94
In the following exercises, find the Maclaurin series for the given function. $$ f(x)=\cos (3 x) $$
Problem 95
In the following exercises, find the Maclaurin series for the given function. $$ f(x)=\ln (x+1) $$
Problem 96
In the following exercises, find the Maclaurin series for the given function. $$ f(x)=\sin x, a=\frac{\pi}{2} $$
