Limit Comparison Test
Consider two series
โ
X
n=1
anand
โ
X
n=1
bnwith positive terms. Suppose that lim
nโโ
an
bn
=C
with 0 < C < โ. Then
1. If
โ
X
n=1
bnconverges, then
โ
X
n=1
anconverges.
2. If
โ
X
n=1
bndiverges, then
โ
X
n=1
andiverges.
USED: When your given series behaves more like a simpler series, when nis large, but you may
not have a direct, obvious, and helpful bound as with the Comparison Test. Also only used for
positive termed series.
USED: LCT does not concern itself with which terms are bigger or smaller. LCT only cares if
the given terms and the comparison terms are about the same in size, as ngets very big. LCT is
the lazy version of comparison, and focuses on the idea that the original series and the (simpler?!)
comparison series share the same convergence behavior.
GOOD FOR: Xpolynomial
polynomial
NOTE: The order of the stack in the limit of an
bn
vs. bn
an
is not so important, because you are
just trying to decide if Cis finite and non-zero. It is often easier to put the given terms in the
numerator and the comparison terms in the denominator.
NOTE: If C= 0 or C=โ, then you either made a mistake in the Limit computation, OR chose
the wrong comparison series to start. Ignore all non-dominant terms as ngrows large, and try
again.
APPROACH:
โขGiven the original series, start by ignoring non-dominant terms and decide what the compar-
ison series will be. Again, this comparison series is usually a p-series or a geometric series.
โขCompare the terms. Compute the Limit of the stack of the given terms over the comparison
terms that are simpler. Justify the limit answer carefully. No guesses. If you use LโH Rule,
you must switch to the related function and the xvariable.
โขAnalyze the comparison series completely.
โขMake a conclusion about convergence for the original series.
