91Ó°ÊÓ

Find the first four elements of the sequence of partial sums \(\left\\{s_{n}\right\\}\) and find a formula for \(s_{n}\) in terms of \(n ;\) also, determine if the infinite series is convergent or divergent, and if it is convergent, find its sum.\(\sum_{n=1}^{+x} \frac{2^{n-1}}{3^{n}}\)

Use a binomial series to find the Maclaurin series for the given function. Determine the radius of convergence of the resulting series.\(f(x)=\frac{x}{\sqrt[3]{1+x^{2}}}\)

A function \(f\) is defined by a power series. In each exercise do the following: (a) Find the radius of convergence of the given power series and the domain of \(f\); (b) write the power series which defines the function \(f^{\prime}\) and find its radius of convergence by using methods of Sec. \(16.7\) (thus verifying Theorem 16.8.1); (c) find the domain of \(f^{\prime}\).\(f(x)=\sum_{n=1}^{+\infty} \frac{x^{2 n-2}}{(2 n-2) !}\)

Integrate term by term from 0 to \(x\) the binomial series for \(\left(1+t^{2}\right)^{-1 / 2}\) to obtain the Maclaurin series for \(\sinh ^{-1} x\). Determine the radius of convergence.

Determine if the given series is convergent or divergent.\(\sum_{k=1}^{+\infty} \frac{n !}{(n+2) !}\)

A function \(f\) is defined by a power series. In each exercise do the following: (a) Find the radius of convergence of the given power series and the domain of \(f\); (b) write the power series which defines the function \(f^{\prime}\) and find its radius of convergence by using methods of Sec. \(16.7\) (thus verifying Theorem 16.8.1); (c) find the domain of \(f^{\prime}\).\(f(x)=\sum_{n=1}^{+\infty} \frac{(x-1)^{n}}{n 3^{n}}\)

Determine if the given sequence is increasing, decreasing, or not monotonic.\(\left\\{\frac{n !}{3^{n}}\right\\}\)

Compute the value of the given integral, accurate to four decimal places, by using series.\(\int_{0}^{1} f(x) d x\), where \(f(x)= \begin{cases}\frac{e^{x}-1}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases}\)

Find the infinite series which is the given sequence of partial sums; also determine if the infinite series is convergent or divergent, and if it is convergent, find its sum.\(\left\\{s_{n}\right\\}=\left\\{\frac{2 n}{3 n+1}\right\\}\)

Determine if the given alternating series is convergent or divergent.\(\sum_{n=1}^{+\infty}(-1)^{n} \frac{n}{2^{n}}\)