%138 Taylor polynomials, interval of convergence
%worksheet

\documentclass{article}
\usepackage{amssymb,amsmath}
\pagestyle{empty}
\begin{document}

 \begin{flushright}
      4/5/13
  \end{flushright}



\begin{center}
{\bf Exam III, Chapter 8 \& Sections 11.7 - 11.11}
\end{center}

\large

4.  The Maclaurin series for the function $\cos x$ is shown below  Carefully show that the
interval of convergence for the series is $-\infty < x < +\infty$.
\[
	1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots =
	\sum_{n=0}^{+\infty} \frac{(-1)^{n}}{(2n)!}x^{2n}
\]

\vfill

\vfill

5.  Find the fifth degree Taylor polynomial for the function\\ $f(x) = \cos x - \sin x$.

\vfill


\vfill

7.  Use an eighth degree Taylor polynomial to estimate
\[
	\int_0^1 \cos (x^2) \, dx
\]

\pagebreak

8.  The Maclaruin series for the funciton $\tan^{-1}x$ is below.  
Carefully show that the interval of convergence for the series is $[-1, 1]$.  
\[
	x - \frac{x^3}{3}+\frac{x^5}{5} - \frac{x^7}{7} + \cdots =
	\sum_{n=0}^{+\infty} \frac{(-1)^{n}}{2n+1}x^{2n+1}
\]

\vfill

\vfill

9.  Use the above to give an example of an infinite series that converges to $\pi$.\\

\end{document}
