Direct comparison test

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.

Comparison test of the first kind

The first comparison test states that if the series

${\displaystyle \sum _{n=1}^{\infty }b_{n}}$ 

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

${\displaystyle |a_{n}|\leq C|b_{n}|}$ 

for sufficiently large n , then the series

${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ 

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

${\displaystyle |a_{n}|\geq |b_{n}|}$ 

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).

Comparison test of the second kind

The second comparison test states that if the series

${\displaystyle \sum _{n=1}^{\infty }b_{n}}$ 

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

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\leq C\,\left|{\frac {b_{n+1}}{b_{n}}}\right|}$ 

for sufficiently large n , then the series

${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ 

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

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\geq \left|{\frac {b_{n+1}}{b_{n}}}\right|}$ 

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.

References

• Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.1) ISBN 0486601536

• Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 2.34) ISBN 0521588073

See also

• Radius of convergence