Limit Comparison Test
Suppose and
are two infinite series.
Suppose also that
r = lim  a _{n} / b _{n} exists and 0 < r < Then converges absolutely if and only if converges absolutely.
This test is more useful than the "direct" comparison test because you do not need to compare the terms of two series too carefully. It is sufficient if the two terms behave similar "in the long run".
Examples 4.2.6:  

Since r = lim  a _{n} / b _{n}  exists, and r is between 0 and infinity there exist constants c and C, 0 < c < C < such that for some positive integer N we have:
c <  a _{n} / b _{n}  < Cif n > N. Assume converges absolutely. From above we have that
c  b _{n}  <  a_{n} for n > N. Hence, converges absolutely by the comparison test.
Assume converges absolutely. From above we have that
a _{n}  < C  b _{n} for n > N. But since the series C also converges absolutely, we can use again the comparison test to see that must converge absolutely.