Chapter 16

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

Prove that the series $$ \sum_{n=0}^{+\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !} $$ represents \(\cos x\) for all values of \(x\).

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

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

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

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

Determine if the given sequence is increasing, decreasing, or not monotonic.\(\left\\{\frac{3 n-1}{4 n+5}\right\\}\)

Prove that the series $$ \sum_{n=0}^{+\infty} \frac{x^{2 n+1}}{(2 n+1) !} $$ represents \(\sinh x\) for all values of \(x\).

Compute the value of the given integral, accurate to four decimal places, by using series.\(\int_{0}^{1 / 3} \frac{d x}{1+x^{4}}\)

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

Find the interval of convergence of the given power series.\(\sum_{n=0}^{+\infty} \frac{x^{n}}{n+1}\)

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

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

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.\(\sum_{n=1}^{+\infty} \frac{2}{(4 n-3)(4 n+1)}\)

Determine if the given sequence is increasing, decreasing, or not monotonic.\(\\{\sin n \pi\\}\)

Use a binomial series to find the Maclaurin series for the given function. Determine the radius of convergence of the resulting series.\(f(x)=\left(4+x^{2}\right)^{-1}\)

Prove that the series $$ \sum_{n=0}^{+\infty} \frac{x^{2 n}}{(2 n) !} $$ represents \(\cosh x\) for all values of \(x\).

Compute the value of the given integral, accurate to four decimal places, by using series.\(\int_{0}^{1} e^{-x^{2}} d x\)

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

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

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

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

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.\(\sum_{n=1}^{+\infty} \ln \frac{n}{n+1}\)

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

Use a binomial series to find the Maclaurin series for the given function. Determine the radius of convergence of the resulting series.\(f(x)=\sqrt[3]{8+x}\)

Compute the value of the given integral, accurate to four decimal places, by using series.\(\int_{0}^{1 / 2} f(x) d x\), where \(f(x)= \begin{cases}\frac{\ln (1+x)}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases}\)

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

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

Determine if the given series is convergent or divergent.\(\sum_{n=1}^{+\infty} \frac{n^{2}}{4 n^{3}+1}\)

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

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.

Compute the value of the given integral, accurate to four decimal places, by using series.\(\int_{0}^{1 / 4} g(x) d x\), where \(g(x)= \begin{cases}\frac{\tan ^{-1} x}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{cases}\)

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

Determine if the given series is convergent or divergent.\(\sum_{n=1}^{+\infty} \frac{1}{\sqrt{n^{2}+4 n}}\)

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

Determine if the given sequence is increasing, decreasing, or not monotonic.\(\left\\{\frac{5^{n}}{1+5^{2 n}}\right\\}\)

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

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

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

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

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

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

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

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.

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

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

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

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