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.
