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Mathematics LibreTexts

Summary of Convergence Tests

  • Page ID
    17116
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    Summary of Convergence Tests

     

    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.