\documentclass{article} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} %\usepackage{ulem} % for strikethrough \sout{...} command %\usepackage{color} %\usepackage{tikz} % for tikz diagrams %\usepackage{wasysym} % \smiley \setlength{\unitlength}{0.1in} \setlength{\oddsidemargin}{-0.375in} \setlength{\evensidemargin}{-0.375in} \setlength{\topmargin}{-0.25in} \setlength{\headheight}{0.0in} \setlength{\headsep}{0.0in} \setlength{\topskip}{0.0in} \setlength{\textheight}{9.75in} \setlength{\textwidth}{7.25in} %\newcommand{\bdot}{\, {\bf \cdot} \,} \newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,} \newcommand{\B}[1]{{\bf {#1}}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\myspace}{\parbox[c][0.5in]{0.5in}{\hfill}} \newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}} \newcommand{\force}{{\noindent $\mbox{\;}$}} \newcommand{\diagquotient}[2]{\displaystyle{{#1 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop #2}}} \begin{document} \pagestyle{empty} \noindent \parbox{2in}{\bf Math 1120 Form A} \hfill {\Large \bf Test \#3} \hfill \parbox{2in}{\bf \hfill Nov. $10^\mathrm{th}$, 2021} \vspace{0.3in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{0.2in} \noindent {\bf \large 1. (22 points)} Improper Integrals \begin{enumerate}[(a)] \item Let $\displaystyle I = \int_{-\infty}^{10} \dfrac{x^2+\cos(x)}{(x-3)^5}\,dx$. Write $I$ as a sum of limits of proper integrals.\\ \force \hfill {\it Note:} {\bf Do not try to evaluate these integrals.} It'll only end in tears. \vfill \item {\bf Compute} the integral $\displaystyle \int_0^2 \dfrac{x}{(x^2-4)^2}\,dx$ \qquad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } (circle your answer). \vfill \vfill \item Use an integral comparison test show that $\displaystyle \int_1^{\infty} \dfrac{1+\sin^2(x)}{x^2+1}$ \qquad {\large Converges \quad / \quad Diverges } (circle your answer). {\it Note:} You should {\it both} write down a comparison {\it and} compute the integral you are comparing with. \vfill \vfill \end{enumerate} \noindent {\bf \large 2. (10 points)} Write down the first 3 terms of each of the following sequences.\\ \force \hfill If the sequence converges, circle ``Converges'' and find its limit. If not, circle ``Diverges''. \vspace{0.2in} \begin{enumerate}[(a)] \item $\displaystyle \left\{ n\cos(n) \large\strut \right\}_{\bf n=0}^\infty$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \quad \begin{tabular}{ccccc} \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} \\ $1^\text{st}$ term & & $2^\text{nd}$ term & & $3^\text{rd}$ term\end{tabular} \vspace{0.5in} \item $\displaystyle \left\{ \dfrac{\ln(k)}{k} \right\}_{\bf k=2}^\infty$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \quad \begin{tabular}{ccccc} \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} \\ $1^\text{st}$ term & & $2^\text{nd}$ term & & $3^\text{rd}$ term\end{tabular} \vspace{0.5in} \end{enumerate} \newpage \noindent {\bf \large 3. (34 points)} Summing Series? \begin{enumerate}[(a)] \item Find the third partial sum of $\displaystyle \sum\limits_{n=1}^\infty \left((2n-1)^2-(2n+1)^2\large\strut\right)$. \qquad (Don't worry about simplifying $S_3$.)\\ \force \hfill Also, if the series converges, circle ``Converges'' and find its sum. Otherwise, circle ``Diverges''. $S_3 \quad =$ \quad \underline{\hspace*{0.75in}} \qquad \qquad $\displaystyle \sum\limits_{n=1}^\infty \left((2n-1)^2-(2n+1)^2\large\strut\right)$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \vfill \item Find the third partial sum of $\displaystyle \sum\limits_{n=0}^\infty \dfrac{(-1)^n 10}{2^{n+1}}$. \qquad (Don't worry about simplifying $S_3$.)\\ \force \hfill Also, if the series converges, circle ``Converges'' and find its sum. Otherwise, circle ``Diverges''. $S_3 \quad =$ \quad \underline{\hspace*{0.75in}} \qquad \qquad $\displaystyle \sum\limits_{n=0}^\infty \dfrac{(-1)^n 10}{2^{n+1}}$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \vfill \item Find the second partial sum. Then use the alternating series test to show this series converges. Finally, use your partial sum to help give bounds for the sum of the series. $$S_2 \quad = \quad \underline{\hspace*{0.75in}} \qquad \qquad \mbox{Bounds:} \quad \underline{\hspace*{1in}} \quad \leq \quad \displaystyle\sum_{k=1}^\infty \dfrac{(-1)^{k+1}}{k+1} \quad \leq \quad \underline{\hspace*{1in}}$$ \vfill \item Use the integral test to show this series converges. Then use integral test results find bounds for its sum. $$\mbox{Bounds:} \quad \underline{\hspace*{1in}} \quad \leq \quad \displaystyle\sum_{k=0}^\infty e^{-2k} \quad \leq \quad \underline{\hspace*{1in}}$$ \vfill \end{enumerate} \newpage \noindent {\bf \large 4. (34 points)} Converges Conditionally, Converges Absolutely, or Diverges?\\ Please circle your answer. Circle the test that you used. And show your work (apply the test). \begin{enumerate}[(a)] \item $\displaystyle{\sum_{n=0}^\infty \frac{n^25^n}{n!}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \item $\displaystyle{\sum_{n=1}^\infty \left(\frac{3n+4}{2n+1}\right)^{\!\! n}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \item $\displaystyle{\sum_{n=1}^\infty \frac{n^2+2}{5n^4-n+1}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \item $\displaystyle{\sum_{n=2}^\infty \frac{(-1)^n}{\ln(n)}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \end{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent \parbox{2in}{\bf Math 1120 Form B} \hfill {\Large \bf Test \#3} \hfill \parbox{2in}{\bf \hfill Nov. $10^\mathrm{th}$, 2021} \vspace{0.3in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{0.2in} \noindent {\bf \large 1. (22 points)} Improper Integrals \begin{enumerate}[(a)] \item Let $\displaystyle I = \int_{-\infty}^{7} \dfrac{x^2+\sin(x)}{(x-2)^3}\,dx$. Write $I$ as a sum of limits of proper integrals.\\ \force \hfill {\it Note:} {\bf Do not try to evaluate these integrals.} It'll only end in tears. \vfill \item {\bf Compute} the integral $\displaystyle \int_0^3 \dfrac{x}{(x^2-9)^2}\,dx$ \qquad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } (circle your answer). \vfill \vfill \item Use an integral comparison test show that $\displaystyle \int_1^{\infty} \dfrac{1+\cos^2(x)}{x^2+1}$ \qquad {\large Converges \quad / \quad Diverges } (circle your answer). {\it Note:} You should {\it both} write down a comparison {\it and} compute the integral you are comparing with. \vfill \vfill \end{enumerate} \noindent {\bf \large 2. (10 points)} Write down the first 3 terms of each of the following sequences.\\ \force \hfill If the sequence converges, circle ``Converges'' and find its limit. If not, circle ``Diverges''. \vspace{0.2in} \begin{enumerate}[(a)] \item $\displaystyle \left\{ \dfrac{\cos(n)}{n+1} \large\strut \right\}_{\bf n=0}^\infty$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \quad \begin{tabular}{ccccc} \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} \\ $1^\text{st}$ term & & $2^\text{nd}$ term & & $3^\text{rd}$ term\end{tabular} \vspace{0.5in} \item $\displaystyle \left\{ (-1)^k\ln(k) \right\}_{\bf k=1}^\infty$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \quad \begin{tabular}{ccccc} \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} & , & \underline{\hspace*{0.75in}} \\ $1^\text{st}$ term & & $2^\text{nd}$ term & & $3^\text{rd}$ term\end{tabular} \vspace{0.5in} \end{enumerate} \newpage \noindent {\bf \large 3. (34 points)} Summing Series? \begin{enumerate}[(a)] \item Find the third partial sum of $\displaystyle \sum\limits_{n=1}^\infty \left(\ln(2n-1)-\ln(2n+1)\large\strut\right)$. \qquad (Don't worry about simplifying $S_3$.)\\ \force \hfill Also, if the series converges, circle ``Converges'' and find its sum. Otherwise, circle ``Diverges''. $S_3 \quad =$ \quad \underline{\hspace*{0.75in}} \qquad \qquad $\displaystyle \sum\limits_{n=1}^\infty \left(\ln(2n-1)-\ln(2n+1)\large\strut\right)$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \vfill \item Find the third partial sum of $\displaystyle \sum\limits_{n=0}^\infty \dfrac{(-1)^n 4}{3^{n+1}}$. \qquad (Don't worry about simplifying $S_3$.)\\ \force \hfill Also, if the series converges, circle ``Converges'' and find its sum. Otherwise, circle ``Diverges''. $S_3 \quad =$ \quad \underline{\hspace*{0.75in}} \qquad \qquad $\displaystyle \sum\limits_{n=0}^\infty \dfrac{(-1)^n 4}{3^{n+1}}$ \quad {\large Converges to \underline{\hspace*{0.5in}} \quad / \quad Diverges } \vfill \item Find the second partial sum. Then use the alternating series test to show this series converges. Finally, use your partial sum to help give bounds for the sum of the series. $$S_2 \quad = \quad \underline{\hspace*{0.75in}} \qquad \qquad \mbox{Bounds:} \quad \underline{\hspace*{1in}} \quad \leq \quad \displaystyle\sum_{k=1}^\infty \dfrac{(-1)^{k}}{k} \quad \leq \quad \underline{\hspace*{1in}}$$ \vfill \item Use the integral test to show this series converges. Then use integral test results find bounds for its sum. $$\mbox{Bounds:} \quad \underline{\hspace*{1in}} \quad \leq \quad \displaystyle\sum_{k=0}^\infty e^{-5k} \quad \leq \quad \underline{\hspace*{1in}}$$ \vfill \end{enumerate} \newpage \noindent {\bf \large 4. (34 points)} Converges Conditionally, Converges Absolutely, or Diverges?\\ Please circle your answer. Circle the test that you used. And show your work (apply the test). \begin{enumerate}[(a)] \item $\displaystyle{\sum_{n=0}^\infty \frac{4^n}{n^2 \cdot n!}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \item $\displaystyle{\sum_{n=1}^\infty \left(\frac{2n+1}{3n+4}\right)^{\!\! n}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \item $\displaystyle{\sum_{n=1}^\infty \frac{n^2+2}{5n^3-n+1}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \item $\displaystyle{\sum_{n=2}^\infty \frac{(-1)^n}{\ln(n)}}$ \qquad \qquad {\large Converges Conditionally / Converges Absolutely / Diverges} \noindent \mbox{} \hspace*{-0.25in} {\small $n^{th}$-term Divergence Test / Comparison Test / Integral Test / Ratio Test / Root Test / Alternating Series Test / Other} \vfill \end{enumerate} \end{document}