91Ó°ÊÓ

Find the sum of the convergent series. $$ 4-2+1-\frac{1}{2}+\cdots $$

Determine the convergence or divergence of the sequence. If the sequence converges, use a symbolic algebra utility to find its limit. $$ a_{n}=\frac{n+1}{n^{2}-3} $$

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty}(-1)^{n} e^{-n} $$

Use a sixth-degree Taylor polynomial centered at for the function \(f\) to obtain the required approximation. Function \(\quad\) Approximation \(f(x)=\sqrt{x}, \quad c=4 \quad f(5)\)

Determine the convergence or divergence of the sequence. If the sequence converges, use a symbolic algebra utility to find its limit. $$ a_{n}=\frac{3^{n}}{4^{n}} $$

Use a sixth-degree Taylor polynomial centered at zero to approximate the integral. Function \(\quad\) Approximation \(f(x)=e^{-x^{2}}\) \(\quad\) \(\int_{0}^{1} e^{-x^{2}} d x\)

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}-\frac{1}{3^{n}}\right) $$

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} 2^{n}}{n !} $$

Apply Taylor’s Theorem to find the power series (centered at ) for the function, and find the radius of convergence. Function \(\quad\) Center \(f(x)=e^{3 x} \quad c=0\)

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. \(f(x)=x^{2}-\ln x-\frac{3}{2}\)