Chapter 10

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{n+1}{n^{2} \sqrt{n}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{\sin n}{n^{2}} $$

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? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} 3^{2 n}(x-2)^{n}}{3 n} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{5^{n}}{4^{n}+3} $$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$0 . \overline{d}=0 . d d d d \ldots, \quad where \quad d \quad is \quad a \quad digit$$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(2,6,10,14,18, \dots\)

Estimate the error if \(\cos t^{2}\) is approximated by \(1-\frac{t^{4}}{2}+\frac{t^{8}}{4 !}\) in the integral \(\int_{0}^{1} \cos t^{2} d t\)

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$x \tan ^{-1} x^{2}$$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=x^{3}-2 x+4, \quad a=2\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{10 n+1}{n(n+1)(n+2)}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{3+n}{5+n} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{2+(-1)^{n}}{1.25^{n}}$$

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? $$ \sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{n} x^{n} $$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 0.0 \overline{6}=0.06666 \ldots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. $$ \frac{5}{1}, \frac{8}{2}, \frac{11}{6}, \frac{14}{24}, \frac{17}{120}, \dots $$

Estimate the error if cos \(\sqrt{t}\) is approximated by \(1-\frac{t}{2}+\frac{t^{2}}{4 !}-\frac{t^{3}}{6 !}\) in the integral \(\int_{0}^{1} \cos \sqrt{t} d t\)

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\sin x \cdot \cos x$$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=2 x^{3}+x^{2}+3 x-8, \quad a=1\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{5 n^{3}-3 n}{n^{2}(n-2)\left(n^{2}+5\right)}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{(-2)^{n+1}}{n+5^{n}} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(-2)^{n}}{3^{n}}$$

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? $$ \sum_{n=1}^{\infty}(\ln n) x^{n} $$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 1 . \overline{414}=1.414414414 \ldots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. $$ \frac{1}{25}, \frac{8}{125}, \frac{27}{625}, \frac{64}{3125}, \frac{125}{15,625}, \ldots $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{1}{2 n-1} $$

In Exercises \(25-28\) , find a polynomial that will approximate \(F(x)\) throughout the given interval with an error of magnitude less than \(10^{-3} .\) \begin{equation} F(x)=\int_{0}^{x} \sin t^{2} d t, \quad[0,1] \end{equation}

In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{3}{n}\right)^{n}$$

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$e^{x}+\frac{1}{1+x}$$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=x^{4}+x^{2}+1, \quad a=-2\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty}\left(\frac{n}{3 n+1}\right)^{n}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1+n}{n^{2}} $$

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? $$ \sum_{n=1}^{\infty} n^{n} x^{n} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{2^{n}}{n+1} $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,0,1,0,1, \ldots\)

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\cos x-\sin x$$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=3 x^{5}-x^{4}+2 x^{3}+x^{2}-2, \quad a=-1\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}+2}}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n+1}(\sqrt[n]{10}) $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{3}{n}\right)^{n}$$

Express each of the numbers in Exercises \(19-26\) as the ratio of two integers. $$ 3 . \overline{142857}=3.142857142857 \ldots $$

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(0,1,1,2,2,3,3,4, \ldots\)

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? $$ \sum_{n=0}^{\infty} n !(x-4)^{n} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{n} $$

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\frac{x}{3} \ln \left(1+x^{2}\right)$$

Find the Taylor series generated by \(f\) at \(x=a.\) \(f(x)=1 / x^{2}, \quad a=1\)

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=3}^{\infty} \frac{1}{\ln (\ln n)}\end{equation}

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} n^{2}(2 / 3)^{n} $$

Determining Convergence or Divergence In Exercises \(17-44,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}}$$

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? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x+2)^{n}}{n 2^{n}} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=2}^{\infty} \frac{\sqrt{n}}{\ln n} $$