Comparison test

Comparison test

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the comparison test, sometimes called the direct comparison test or CQT (in contrast with the related limit comparison test
Limit comparison test
In mathematics, the limit comparison test is a method of testing for the convergence of an infinite series.- Statement :...

) is a criterion for convergence or divergence
Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....

 of a series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 whose terms are real or complex numbers. The test determines convergence by comparing the terms of the series in question with those of a series whose convergence properties are known.

Statement


The comparison test states that if the series
is an absolutely convergent series and for sufficiently large n , then the series
converges absolutely. In this case b is said to "dominate" a. If the series
is divergent and
for sufficiently large n , then the series
also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an  alternate in sign).

Proof


Let . Let the partial sums of these series be and respectively i.e.
converges as . Denote its limit as . We then have
which gives us
This shows that is a bounded monotonic sequence and must converge to a limit.