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| Series or Test | Conclusions | Comments |
|---|---|---|
| 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. | ||
| 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. |
