91Ó°ÊÓ

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} \frac{k}{k^{3}+k+1}\)

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\frac{1}{\sqrt{1+x}}, \quad n=2 $$

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt[3]{8.3} $$

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=(1-x)^{\frac{2}{3}}, \quad n=3 $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=1}^{\infty} \frac{2}{3^{k}+1}\)

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \sqrt{103} $$

Use your knowledge of the binomial series to find the \(n\) th degree Taylor polynomial for \(f(x)\) about \(x=0 .\) Give the radius of convergence of the corresponding Maclaurin series. One of these "series" converges for all \(x\). $$ f(x)=\sqrt[3]{1+x^{2}}, \quad n=5 $$

Write the given integral as a power series. \(\int \frac{1}{1+x^{5}} d x\)

Use a second degree Taylor polynomial centered appropriately to approximate the expression given. $$ \tan ^{-1}(0.75) $$

Determine whether the series converges or diverges. It is possible to solve Problems 4 through 19 without the Limit Comparison, Ratio, and Root Tests. \(\sum_{k=2}^{\infty} \frac{5}{k-0.5}\)