Summary of Convergence Tests
- Page ID
- 17116
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| Series or Test | Conclusions | Comments |
|---|---|---|
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Divergence Test For any series \( \sum^∞_{n=1}a_n\), evaluate \( \lim_{n→∞}a_n\). |
If \( \lim_{n→∞}a_n=0\), the test is inconclusive. | This test cannot prove convergence of a series. |
| If \( \lim_{n→∞}a_n≠0\), the series diverges. | ||
| Geometric Series \(\sum^∞_{n=1}ar^{n−1}\) | If \( |r|<1\), the series converges to \( a/(1−r)\). | Any geometric series can be reindexed to be written in the form \( a+ar+ar^2+⋯\), where \( a\) is the initial term and r is the ratio. |
| If \( |r|≥1,\) the series diverges. | ||
|
p-Series \( \sum^∞_{n=1}\frac{1}{n^p}\) |
If \( p>1\), the series converges. | For \( p=1\), we have the harmonic series \( \sum^∞_{n=1}1/n\). |
| If \( p≤1\), the series diverges. | ||
|
Comparison Test For \( \sum^∞_{n=1}a_n \) with nonnegative terms, compare with a known series \( \sum^∞_{n=1}b_n\). |
If \( a_n≤b_n\) for all \( n≥N\) and \( \sum^∞_{n=1}b_n\) converges, then \( \sum^∞_{n=1}a_n\) converges. | Typically used for a series similar to a geometric or \( p\)-series. It can sometimes be difficult to find an appropriate series. |
| If \( a_n≥b_n\) for all \( n≥N\) and \( \sum^∞_{n=1}b_n\) diverges, then \( \sum^∞_{n=1}a_n\) diverges. | ||
|
Limit Comparison Test For \( \sum^∞_{n=1}a_n\) with positive terms, compare with a series \( \sum^∞_{n=1}b_n\) by evaluating \( L=\lim_{n→∞}\frac{a_n}{b_n}.\) |
If \( L\) is a real number and \( L≠0\), then \( \sum^∞_{n=1}a_n\) and \( \sum^∞_{n=1}b_n\) both converge or both diverge. | Typically used for a series similar to a geometric or \( p\)-series. Often easier to apply than the comparison test. |
| If \( L=0\) and \( \sum^∞_{n=1}b_n\) converges, then \( \sum^∞_{n=1}a_n\) converges. | ||
| If \( L=∞\) and \( \sum^∞_{n=1}b_n\) diverges, then \( \sum^∞_{n=1}a_n\) diverges. | ||
|
Integral Test If there exists a positive, continuous, decreasing function \( f\) such that \( a_n=f(n)\) for all \( n≥N\), evaluate \( ∫^∞_Nf(x)dx.\) |
\( ∫^∞_Nf(x)dx\) and \( \sum^∞_{n=1}a_n\) both converge or both diverge. | Limited to those series for which the corresponding function f can be easily integrated. |
|
Alternating Series \( \sum^∞_{n=1}(−1)^{n+1}b_n\) or \( \sum^∞_{n=1}(−1)^nb_n\) |
If \( b_{n+1}≤b_n\) for all \( n≥1\) and \( b_n→0\), then the series converges. | Only applies to alternating series. |
|
Ratio Test For any series \( \sum^∞_{n=1}a_n\) with nonzero terms, let \( ρ=\lim_{n→∞}∣\frac{a_{n+1}}{a_n}∣\) |
If \( 0≤ρ<1\), the series converges absolutely. |
Often used for series involving factorials or exponentials.
|
| If \( ρ>1\) or \( ρ=∞\), the series diverges. | ||
| If \( ρ=1\), the test is inconclusive. | ||
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Root Test For any series \( \sum^∞_{n=1}a_n\), let \( ρ=\lim_{n→∞}\sqrt[n]{|a_n|}\). |
If \( 0≤ρ<1\), the series converges absolutely. | Often used for series where \( |a_n|=b^n_n\). |
| If \( ρ>1\) or \( ρ=∞\), the series diverges. | ||
| If \( ρ=1\), the test is inconclusive. |