We compare infinite series to each other using limits.
- If then either both series converge, or they both diverge.
- If and converges, then converges.
- If and diverges, then diverges.
This theorem should make intuitive sense.
- If then we have for large , so the behavior of the respective series should be the same.
- If then should be way less than . So if converges, should also converge by the comparison test.
- If , then should be way greater than . So if diverges, should also diverge by the comparison test.
The way we actually use this in practice still involves some creativity: we have to decide on a ‘‘similar’’ series for which we know the convergence properties. However, unlike the comparison test, we can just mechanically take a limit of the ratio of our guess with our original series, instead of having to ‘‘get our hands dirty’’ with inequalities.
Since is convergent by the -series test with , then the limit comparison test applies, and must also converge.
If the limit comparison test is easier to use than the comparison test, why do we even have the comparison test? Sometimes, the comparison test is actually more powerful. The next example illustrates this idea.
Let’s pause another moment to consider the task of choosing a test to use when analyzing a series for convergence or divergence. Take a few minutes to make a list of all the tests we know so far, and the best situations in which to use each of them.