Limit comparison test
Encyclopedia
In mathematics
, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
and 
with 
for all 
.
Then if
with 
then either both series converge or both series diverge.
we know that for all 
exists an 
such that for all 
we have that 
, or what is the same
As
we can choose 
such that 
is positive.
So
and by the direct comparison test
, if
converges then so does 
.
Similarly
, so if 
converges, again by the direct comparison test, so does 
.
That is both series converge or both series diverge.
converges. For this we compare with the convergent series 
.
As
we have that the original series also converges.
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 limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
Statement
Suppose that we have two series and with for all .Then if
with then either both series converge or both series diverge.Proof
Because we know that for all exists an such that for all we have that , or what is the same
As
we can choose such that is positive.So
and by the direct comparison testComparison test
In mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...
, if
converges then so does .Similarly
, so if converges, again by the direct comparison test, so does .That is both series converge or both series diverge.
Example
We want to determine if the series converges. For this we compare with the convergent series .As
we have that the original series also converges.