Comparison test

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In mathematics, the comparison test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. It determines convergence by comparing the terms of the series in question with those of a series whose convergence properties are known. There are two versions of the comparison test.

1 Comparison test of the first kind
2 Comparison test of the second kind
3 References
4 See also

[edit] Comparison test of the first kind

The first comparison test states that if the series

$\sum_{n=1}^\infty b_n$

is an absolutely convergent series and there exists a real number C independent of n such that

$|a_n|\le C|b_n|$

for sufficiently large n , then the series

$\sum_{n=1}^\infty a_n$

converges absolutely. In this case b is said to "dominate" a. If the series ∑|b_n | is divergent and

$|a_n|\ge |b_n|$

for sufficiently large n , then the series ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).

[edit] Comparison test of the second kind

The second comparison test states that if the series

$\sum_{n=1}^\infty b_n$

is an absolutely convergent series and there exists a real number C independent of n such that

$\left|\frac{a_{n+1}}{a_n}\right|\le C\,\left|\frac{b_{n+1}}{b_n}\right|$

for sufficiently large n , then the series

$\sum_{n=1}^\infty a_n$

converges absolutely. If the series ∑|b_n | is divergent and

$\left|\frac{a_{n+1}}{a_n}\right|\ge \left|\frac{b_{n+1}}{b_n}\right|$

for sufficiently large n , then the series ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).

This is based upon Jean de Rond d'Alembert's ratio test.