91Ó°ÊÓ

Use a graphing utility to approximate all the real zeros of the function by Newton’s Method. Graph the function to make the initial estimate of a zero. \(f(x)=\frac{1}{4} x^{3}-3 x^{2}+\frac{3}{4} x-2\)

Determine the convergence or divergence of the sequence. If the sequence converges, find its limit. $$ a_{n}=\frac{n^{2}-25}{n+5} $$

Determine the convergence or divergence of the \(p\)-series. \(1+\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\cdots\)

Use a symbolic algebra utility to find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^{n}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots $$

Use a symbolic differentiation utility to find the fourth-degree Taylor polynomial (centered at zero). \(f(x)=\frac{1}{\sqrt[3]{x+1}}\)

Find the radius of convergence of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n+1}(x-1)^{n+1}}{n+1} $$

Use a symbolic algebra utility to find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 2\left(\frac{2}{3}\right)^{n}=2+\frac{4}{3}+\frac{8}{9}+\frac{16}{27}+\cdots $$

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

Use a symbolic differentiation utility to find the fourth-degree Taylor polynomial (centered at zero). \(f(x)=x e^{x}\)

Determine the convergence or divergence of the sequence. If the sequence converges, find its limit. $$ a_{n}=\frac{n+2}{n^{2}+1} $$