91Ó°ÊÓ

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\left\\{1,-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \ldots\right\\}$$

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=\frac{\ln x}{x}, \quad a=1$$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n^{2}}$$

\(3-8=\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\sqrt{n^{3}+2}} $$

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n !}$$

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{1}{x+10} $$

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\left\\{\frac{1}{2},-\frac{4}{3}, \frac{9}{4},-\frac{16}{5}, \frac{25}{6}, \ldots\right\\}$$

Find the Taylor polynomial \(T_{3}(x)\) for the function \(f\) centered at the number \(a .\) Graph \(f\) and \(T_{3}\) on the same screen. $$f(x)=x e^{-2 x}, \quad a=0$$

\(3-8=\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{3 n-1}{2 n+1} $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$