91Ó°ÊÓ

Determine the degree of the Taylor polynomial centered at \(c\) required to approximate \(f\) in the given interval to an accuracy of \(\pm 0.001\). Function \(\quad\) Interval \(f(x)=e^{x}, \quad c=1\) \(\quad\) \([0,2]\)

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{n 5^{n}}{n !} $$

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=1}^{\infty} \frac{n+10}{10 n+1} $$

Apply Taylor’s Theorem to find the power series (centered at ) for the function, and find the radius of convergence. Function \(\quad\) Center \(f(x)=\sqrt{x} \quad c=4\)

Apply Newton’s Method using the indicated initial estimate. Then explain why the method fails. \(y=4 x^{3}-12 x^{2}+12 x-3, \quad x_{1}=\frac{3}{2}\)

Determine the degree of the Taylor polynomial centered at \(c\) required to approximate \(f\) in the given interval to an accuracy of \(\pm 0.001\). Function \(\quad\) Interval \(f(x)=\frac{1}{x}, \quad c=1\) \(\quad\) \(\left[1, \frac{3}{2}\right]\)

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2 n !}{n^{5}} $$

Determine the convergence or divergence of the sequence. If the sequence converges, use a symbolic algebra utility to find its limit. $$ a_{n}=(-1)^{n} \frac{n}{n^{2}+1} $$

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=1}^{\infty} \frac{n !+1}{n !} $$

Determine the maximum error guaranteed by Taylor's Theorem with Remainder when the fifth-degree Taylor polynomial is used to approximate \(f\) in the given interval. \(f(x)=e^{-x}, \quad[0,1], \quad\) centered at 0