91Ó°ÊÓ

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

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

Obtain the Maclaurin series for the hyperbolic sine function by differentiating the Maclaurin series for the hyperbolic cosine function. Also differentiate the Maclaurin series for the hyperbolic sine function to obtain the one for the hyperbolic cosine function.

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

Find the interval of convergence of the given power series.\(\sum_{n=1}^{+\infty} \frac{(x+2)^{n}}{(n+1) 2^{n}}\)

Find a power-series representation for \(\tanh ^{-1} x\) by integrating term by term from 0 to \(x\) a power-series representation for \(\left(1-t^{2}\right)^{-1}\)

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

If \(s_{k}\) is the \(k\) th partial sum of the harmonic series, prove that $$ \ln (k+1)

If \(\sum_{n=1}^{+\infty} a_{n}\) and \(\sum_{n=1}^{+\infty} b_{n}\) are two convergent series of positive terms, use the Limit Comparison Test to prove that the series \(\sum_{n=1}^{+\infty} a_{n} b_{n}\) is also convergent.

Express the given nonterminating repeating decimal as a common fraction.\(1.234234234 \ldots\)