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.
Suppose that we have two series
, then either both series converge or both series diverge.
there is a positive integer
to be sufficiently small such that
and by the direct comparison test, if
converges then so does
diverges, again by the direct comparison test, so does
That is, both series converge or both series diverge.
We want to determine if the series
converges.
For this we compare it with the convergent series
we have that the original series also converges.
One can state a one-sided comparison test by using limit superior.
converges, necessarily
converges.
for all natural numbers
does not exist, so we cannot apply the standard comparison test.
converges, the one-sided comparison test implies that
converges, then necessarily
lim inf
The essential content here is that in some sense the numbers
are larger than the numbers
be analytic in the unit disc
and have image of finite area.
By Parseval's formula the area of the image of
Therefore, by the converse of the comparison test, we have
lim inf
lim inf
lim inf