P-Series and Ratio Tests
Recall that the geometric series is a series where the general term is a constant multiple of rn where r is a constant and n is the index of summation. We we will look at exponents where the constant is the power and the base is the index.
is a P-series with p = -1/2. Recall that roots are just 1/2 powers and pulling up from the denominator is just changing the sign of the exponent.
is a P-series with p = 1. This is just the harmonic series.
is not a P-series since the index is in the exponent. In fact the above series is a geometric series with r = 1/2.
The following test tells us when the p-series converges.
The harmonic series
diverges by the P-series test since
p = 1 < 1
p = 2 > 1
If a series converges then the terms must approach zero as n gets large. However, we have seen that even if the terms approach zero, the series may still diverge. For example, the harmonic series diverges, however the terms do not approach zero fast enough for convergence. The next theorem gives us a method that will often detect whether the terms approach zero fast enough.
The ratio test is especially useful when powers or factorials appear in the general term. For powers, we have
For factorials, note for example