Chapter 8

In Exercises \(1-10\), use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{2 x}, \quad c=0\)

In Exercises \(1-34\), determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}\)

In Exercises 1-12, use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{2 n^{2}+1}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{x^{n}}{n+1} $$

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+2} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{1}{n^{4}} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{n+1}{2 n-1}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{-3 x}, \quad c=0\)

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n \sqrt{n}}\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+2 n}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=1}^{\infty}(-1)^{n-1} n x^{n} $$

Find the \(n\) th partial sum \(S_{n}\) of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. \(\sum_{n=1}^{\infty}\left(\frac{1}{2 n+3}-\frac{1}{2 n+1}\right)\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} n}{3 n-1} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{3}{2 n-1} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{(-1)^{n+1} 2^{n}}{n+1}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{x}, \quad c=2\)

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

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{\sqrt{n}} $$

Find the \(n\) th partial sum \(S_{n}\) of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. \(\sum_{n=1}^{\infty} \frac{4}{(2 n+3)(2 n+5)}\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} e^{-n} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\sin \frac{n \pi}{2}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{-2 x}, \quad c=3\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n^{2 / 3}-1}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} $$

Find the \(n\) th partial sum \(S_{n}\) of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. \(\sum_{n=1}^{\infty}\left(\frac{-8}{4 n^{2}+4 n-3}\right)\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1} n^{2}}{2 n^{2}-1} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} n e^{-n} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{1 \cdot 3 \cdot 5 \cdots \cdot(2 n-1)}{n !}\)

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-2)^{n}}{n !}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=\sin 2 x, \quad c=0\)

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n+1}\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2}-1}}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{(2 x)^{n}}{n !} $$

Find the \(n\) th partial sum \(S_{n}\) of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. \(\sum_{n=2}^{\infty}\left(\frac{1}{\ln n}-\frac{1}{\ln (n+1)}\right)\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\frac{1}{17}+\frac{1}{26}+\cdots $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{2^{n}}{(2 n) !}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=\sin x, \quad c=\frac{\pi}{4}\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=0}^{\infty} \frac{1}{\sqrt{n^{3}+1}}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n}}{n \cdot 3^{n}} $$

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{\sqrt{n^{2}+1}} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{15}+\frac{1}{19}+\cdots $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{1}=2, \quad a_{n+1}=3 a_{n}+1\)

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n^{2}}{n^{2}+3}\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=0}^{\infty} \frac{2^{n}}{3^{n}+1}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=1}^{\infty}(n x)^{n} $$

In Exercises \(7-14\), determine whether the geometric series converges or diverges. If it converges, find its sum. \(4+\frac{8}{3}+\frac{16}{9}+\frac{32}{27}+\cdots\)

Determine whether the series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{(-1)^{n-1} \sqrt{n+1}}{n-1} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{n}{\left(n^{2}+1\right)^{3 / 2}} $$