91Ó°ÊÓ

Use the binomial series to find the power series representation of the function. Then find the radius of convergence of the series. \(f(x)=\sqrt{1-x^{2}}\)

Find an approximation of the sum of the series accurate to two decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{3}} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\tanh n\)

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\sin ^{2} n}{n \sqrt{n+1}}\)

Use the binomial series to find the power series representation of the function. Then find the radius of convergence of the series. \(f(x)=\frac{1}{\sqrt[3]{8+x}}\)

Determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{\tan ^{-1} n}{n^{3}+1}\)

Determine whether the given series converges or diverges. If it converges, find its sum. \(\sum_{n=0}^{\infty} \frac{3^{n+1}}{5^{n}}\)

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\frac{\ln n^{2}}{\sqrt{n}}\)

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n}}{3 \cdot 5 \cdot 7 \cdots(2 n+1)}\)

A Bessel Function The function \(J_{0}\) defined by $$ J_{0}(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{2^{2 n}(n !)^{2}} $$ is called the Bessel function of order \(0 .\) a. What is the domain of \(J_{0}\) ? b. Plot the graph of \(J_{0}\) in the viewing window \([-10,10] \times[-0.5,1.2]\), and plot the graphs of \(S_{n}(x)\) for \(n=0,1,2,3\), and 4 in the viewing window \([-8,8] \times[-2,2]\).