91Ó°ÊÓ

Find an approximation of the sum of the series accurate to two decimal places. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} $$

Find an approximation of the sum of the series accurate to two decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(n+1)}{2^{n}} $$

A Bessel Function The function \(J_{1}\) defined by $$ J_{1}(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{n !(n+1) ! 2^{2 n+1}} $$ is called the Bessel function of order 1 . What is its domain?

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{4 \cdot 7 \cdot 10 \cdot \cdots \cdot(3 n+1)}{4^{n}(n+1) !}\)

Determine whether the given series converges or diverges. If it converges, find its sum. \(\sum_{n=0}^{\infty} \frac{(-3)^{n}}{2^{n+1}}\)

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

Use the binomial series to find the power series representation of the function. Then find the radius of convergence of the series. \(f(x)=(1-x)^{3 / 5}\)

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\frac{2^{n}}{3^{n}+1}\)

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges or diverges. If it converges, find its limit. \(a_{n}=\frac{2^{n}+1}{e^{n}}\)

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