Previous: Example: Integral Test with a Logarithm
Next: Videos on the Integral Test
Problem
For what values of p does the infinite series
converge?
Complete Solution
Step (1): Consider p > 0 and p ≠ 1
When p > 0 and p ≠ 1, the function
is continuous, decreasing, and positive when x is in the interval [1,∞). Using the integral test,
Therefore, the infinite series converges when p > 1, and diverges when p is in the interval (0,1).
Step (2): Consider p ≤ 0 and p = 1
If p=1, then we have the harmonic series
which we know diverges.
If p ≤ 0, the infinite series diverges (by the divergence test).
Therefore, the given series only converges for p > 1.
The p-Series
The result of this example can be summarized as follows.
| The p-Series |
|---|
|
The p-series
is convergent if p > 1 and divergent if p ≤ 1. |
Much like a geometric series, we can use this result to determine whether a given infinite series converges by inspection. For example, the infinite series
diverges because it is a p-series with p equal to 1/2 (you may want to let u=(1+k) to see this).
Previous: Example: Integral Test with a Logarithm
Next: Videos on the Integral Test
