The Ratio and Root Tests
Now I want to bring out with two tests that will tell us about absolute convergence or (unconditional) divergence of an infinite series . As such they’ll tell us nothing about conditionally convergent series.
First is the ratio test. We take the ratio of one term in the series to the previous one and define the limits superior and inferior
Now if then the series converges absolutely. If then the series diverges. But if the test fails and we get no result.
In the first case, pick to be a number so that . Then there is some so that is an upper bound for the sequence of ratios past . For large enough , this means
On the other hand, if then eventually , so the terms of the series are getting bigger and bigger and bigger. But this would throw a monkey wrench into Cauchy’s condition for convergence of the series.
As for the root test, we will consider the sequence and define
If then the series converges absolutely. If then the series diverges. And if the test is inconclusive.
In the first case, as we did for the ratio test, pick so that . Then above some we have and the comparison test works straight away. On the other hand, if then infinitely often, and Cauchy’s criterion falls apart again.