91Ó°ÊÓ

Use Newton’s Method to obtain a general rule for approximating the indicated radical. \(\sqrt[n]{a}\left[\text {Hint}: \text { Consider } f(x)=x^{n}-a .\right]\)

Verify that the Ratio Test is inconclusive for the \(p\)-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}\)

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=0}^{\infty}(1.075)^{n} $$

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) $$ -1,4,9,14, \dots $$

Find the radius of convergence of (a) \(f(x),(b) f^{\prime}(x),(c) f^{\prime \prime}(x),\) and \((d) \int f(x) d x\) $$ f(x)=\sum_{n=0}^{\infty}\left(\frac{x}{2}\right)^{n} $$

Find the radius of convergence of (a) \(f(x),(b) f^{\prime}(x),(c) f^{\prime \prime}(x),\) and \((d) \int f(x) d x\) $$ f(x)=\sum_{n=1}^{\infty} \frac{x^{n}}{n 5^{n}} $$

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) $$ 1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \dots $$

Verify that the Ratio Test is inconclusive for the \(p\)-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{1 / 2}}\)

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{100} $$

Find the radius of convergence of (a) \(f(x),(b) f^{\prime}(x),(c) f^{\prime \prime}(x),\) and \((d) \int f(x) d x\) $$ f(x)=\sum_{n=0}^{\infty} \frac{(x+1)^{n+1}}{n+1} $$