Integral Test Use
- Page ID
- 90454
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)You will be shown a graph and the result of integrating from 1 to \(\infty\). Your task is to decide whether the integral test gives convergence or divergence of the corresponding sum or if the integral test cannot be used. Click on the Start button to begin.
\(\int_1^{\infty}e^{-x}dx=e^1, \displaystyle \sum^∞_{n=1}e^{-x}\)
\(\int_1^{\infty}\frac{1}{\sqrt{x}}dx=\infty, \displaystyle \sum^∞_{n=1}\frac{1}{\sqrt{n}}\)
\(\int_1^{\infty}\frac{\sin(4x)}{x}dx\sim-0.19, \displaystyle \sum^∞_{n=1}\frac{\sin(4n)}{n}\)
\(\int_1^{\infty}(x+1)^\frac{-3}{2}dx=\sqrt{2}, \displaystyle \sum^∞_{n=1}(n+1)^\frac{-3}{2}\)
\(\int_1^{\infty}\frac{1}{(x+1)\ln(x+1)}dx=\infty, \displaystyle \sum^∞_{n=1}\frac{1}{(n+1)\ln(n+1)}\)
\(\int_1^{\infty}\frac{x}{x+1}dx=\infty, \displaystyle \sum^∞_{n=1}\frac{n}{n+1}\)
\(\int_1^{\infty}\frac{x}{e^{x^2}}dx=\frac{1}{2e}, \displaystyle \sum^∞_{n=1}\frac{n}{e^{n^2}}\)
\(\int_1^{\infty}\frac{1}{\sqrt{1+x^2}}dx=\infty, \displaystyle \sum^∞_{n=1}\frac{1}{\sqrt{1+n^2}}\)
\(\int_1^{\infty}\frac{\cos(3x)}{e^x}dx\sim0.052, \displaystyle \sum^∞_{n=1}\frac{\cos(3n)}{e^n}\)
\(\int_1^{\infty}\frac{1}{1+x^2}dx=\frac{\pi}{4}, \displaystyle \sum^∞_{n=1}\frac{1}{1+n^2}\)
\(\int_1^{\infty}\frac{x}{1+x^2}dx=\infty, \displaystyle \sum^∞_{n=1}\frac{n}{1+n^2}\)
\(\int_1^{\infty}\frac{1+2\cos(2x)}{x+\sin(2x)}dx=\infty, \displaystyle \sum^∞_{n=1}\frac{1+2\cos(2n)}{n+\sin(2n)}\)
Diverges Converges Integral Test Cannot be Used