Chapter 10

Find the first four terms of the binomial series for the functions in Exercises \(1-10 .\) \((1+x)^{1 / 2}\)

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=e^{2 x}, \quad a=0\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\sqrt{n}} $$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty} x^{n}$$

In Exercises \(1-8,\) use the Ratio Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$$

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n^{2}+30}\end{equation}

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}} $$

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ 2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n-1}}+\cdots $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{1-n}{n^{2}} $$

Use substitution (as in Example 4) to find the Taylor series at \(x=0\) of the functions in Exercises \(1-10\) . $$e^{-5 x}$$

Find the first four terms of the binomial series for the functions. \begin{equation} (1+x)^{1 / 3} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\sin x, \quad a=0\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{3 / 2}} $$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty}(x+5)^{n}$$

In Exercises \(1-8,\) use the Ratio Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{n+2}{3^{n}}$$

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{n-1}{n^{4}+2}\end{equation}

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n^{0.2}} $$

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ \frac{9}{100}+\frac{9}{100^{2}}+\frac{9}{100^{3}}+\cdots+\frac{9}{100^{n}}+\cdots $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{1}{n !} $$

Find the first four terms of the binomial series for the functions. \begin{equation} (1-x)^{-3} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\ln x, \quad a=1\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n 3^{n}} $$

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty} \frac{(n-1) !}{(n+1)^{2}}$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty}(-1)^{n}(4 x+1)^{n} $$

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}-1}\end{equation}

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} $$

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+(-1)^{n-1} \frac{1}{2^{n-1}}+\cdots $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{(-1)^{n+1}}{2 n-1} $$

Use substitution (as in Example 4) to find the Taylor series at \(x=0\) of the functions in Exercises \(1-10\) . $$5 \sin (-x)$$

Find the first four terms of the binomial series for the functions. \begin{equation} (1-2 x)^{1 / 2} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\ln (1+x), \quad a=0\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=2}^{\infty}(-1)^{n} \frac{4}{(\ln n)^{2}} $$

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty} \frac{2^{n+1}}{n 3^{n-1}}$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty} \frac{(3 x-2)^{n}}{n} $$

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=2}^{\infty} \frac{n+2}{n^{2}-n}\end{equation}

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n+4} $$

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ 1-2+4-8+\cdots+(-1)^{n-1} 2^{n-1}+\cdots $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=2+(-1)^{n} $$

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1+\frac{x}{2}\right)^{-2} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=1 / x, \quad a=2\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n^{2}+1} $$

In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}}$$

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{(x-2)^{n}}{10^{n}} $$

Use the Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{\cos ^{2} n}{n^{3 / 2}}\end{equation}

Use the Integral Test to determine if the series in Exercises \(1-10\) converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$ \sum_{n=1}^{\infty} e^{-2 n} $$

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ \frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots+\frac{1}{(n+1)(n+2)}+\cdots $$

Each of Exercises \(1-6\) gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4} .\) $$ a_{n}=\frac{2^{n}}{2^{n+1}} $$

Find the first four terms of the binomial series for the functions. \begin{equation} \left(1-\frac{x}{3}\right)^{4} \end{equation}

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=1 /(x+2), \quad a=0\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}+5}{n^{2}+4} $$