91Ó°ÊÓ

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.\(\left\\{s_{n}\right\\}=\left\\{\frac{2 n}{3 n+1}\right\\}\)

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 \(f^{\prime}\).\(f(x)=\sum_{n=2}^{+\infty}(-1)^{n} \frac{(x-3)^{n}}{n(n-1)}\)

Determine if the sequence is convergent or divergent. If the sequence converges, find its limit.\(\left\\{\frac{3 n^{3}+1}{2 n^{2}+n}\right\\}\)

Find the interval of convergence of the given power series.\(\sum_{n=1}^{+\infty}(-1)^{n+1} \frac{(n+1) x}{n !}\)

Determine if the given series is convergent or divergent.\(\sum_{n=1}^{+\infty} \frac{1}{\sqrt{n^{3}+1}}\)

Determine if the sequence is convergent or divergent. If the sequence converges, find its limit.\(\left\\{\frac{\ln n}{n^{2}}\right\\}\)

Find the interval of convergence of the given power series.\(\sum_{n=1}^{+\infty}(-1)^{n+1} \frac{(x-1)^{n}}{n}\)

Determine if the given series is convergent or divergent.\(\sum_{n=1}^{+\infty} \frac{n}{5 n^{2}+3}\)

Determine if the given sequence is increasing, decreasing, or not monotonic.\(\left\\{n^{2}+(-1)^{n} n\right\\}\)

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