One of the most important and useful tests for convergence is the P-Series Test.
Why?
Property of the p-Series Test
Because of it’s simplicity and the prominent role it will play in determining convergence for other types of series, such as the the Integral Test and the Comparison Test which we will learn in future lessons.
Here’s a helpful hint… we’re looking for a variable raised to a number!
And the variable must be a fraction!
So, what’s the right fraction?
Well, the test stipulates that we must have the fraction 1 over n all raised to an exponent. And if this is so, then we can determine whether the series will converge or diverge based on the value of the exponent.
Sadly, many textbooks do not fully explain this important feature regarding the fraction, as indicated in blue in the graphic above. But thankfully now that you know what to look for, you’re ready to use this incredible test to your advantage.
Additionally, did you know that the harmonic series, is just a p-series in disguise?
As Oregon State, nicely explains, if our exponent value is one (i.e., p = 1) then the result is a special case of the p-series called the harmonic series which is a divergent infinite series.
Together, we will work through several examples of how to create the just-right fraction, and identify the exponent necessary to determine convergence.
P Series Test Video
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