Here's a full statement of the test:
Suppose \sum_{n=1}^{\infty}a_{n} and \sum_{n=1}^{\infty}b_{n} are two series with positive terms (that is, a_{n} > 0 and b_{n} > 0 for all n). Let r_{n}=a_{n}/b_{n}. Then:
a) If \sum_{n=1}^{\infty}b_{n} converges and lim_{n->\infty}r_{n} exists, then \sum_{n=1}^{\infty}a_{n} converges.
b) If \sum_{n=1}^{\infty}b_{n} diverges and lim_{n->\infty}r_{n}>0 or \lim_{n->\infty}r_{n}=\infty, then \sum_{n=1}^{\infty}a_{n} diverges.
As an simple example, suppose you wish to know whether the series \sum_{n=1}^{\infty}5/(2n^2-1) converges or not. This series is somewhat similar to the p-series \sum_{n=1}^{\infty}1/n^2, which is known to converge. Let a_{n}=5/(2n^2-1) and b_{n}=1/n^2 so that r_{n}=(5n^2)/(2n^2-1). Since r_{n}-> 5/2 as n->\infty, it follows from the limit comparison test that \sum_{n=1}^{\infty}5/(2n^2-1) converges as well. If you are familiar with the "regular" comparison test, note that the limit comparison test is a bit simpler to use for this example.
