91影视

In the following exercises, express each series as a rational function. \(\sum_{n=1}^{\infty} \frac{1}{x^{2 n}}\)

Use \((1+x)^{1 / 3}=1+\frac{1}{3} x-\frac{1}{9} x^{2}+\frac{5}{81} x^{3}-\frac{10}{243} x^{4}+\cdots\) with \(x=1\) to approximate \(2^{1 / 3}\)

In the following exercises, find the Taylor series of the given function centered at the indicated point. $$ x^{4} \text { at } a=-1 $$

Find the radius of convergence \(R\) and interval of convergence for \(\sum a_{n} x^{n}\) with the given coefficients \(a_{n}\). $$ \sum_{n=1}^{\infty} \frac{10^{n} x^{n}}{n !} $$

Use the approximation \((1-x)^{2 / 3}=1-\frac{2 x}{3}-\frac{x^{2}}{9}-\frac{4 x^{3}}{81}-\frac{7 x^{4}}{243}-\frac{14 x^{5}}{729}+\cdots\) for \(|x|<1\) to approximate \(2^{1 / 3}=2.2^{-2 / 3}\)

In the following exercises, express each series as a rational function. \(\sum_{n=1}^{\infty} \frac{1}{(x-3)^{2 n-1}}\)

In the following exercises, find the Taylor series of the given function centered at the indicated point. $$ 1+x+x^{2}+x^{3} \text { at } a=-1 $$

Find the 25 th derivative of \(f(x)=\left(1+x^{2}\right)^{13}\) at \(x=0\).

In the following exercises, express each series as a rational function. \(\sum_{n=1}^{\infty}\left(\frac{1}{(x-3)^{2 n-1}}-\frac{1}{(x-2)^{2 n-1}}\right)\)

Find the radius of convergence \(R\) and interval of convergence for \(\sum a_{n} x^{n}\) with the given coefficients \(a_{n}\). $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{x^{n}}{\ln (2 n)} $$