What is a p-series?
is known as a p-series, where p is a positive constant.
P-Series Test
In other words, the series converges if p is greater than 1. The series diverges if p is less than or equal to 1. The p-series where p = 1 is the harmonic series. Pretty easy right?
Examples
Worked Solutions
Since the exponent must be greater than 1, we can set up an inequality representing the system.
This problem is a bit tricker since we also have to use the integral test with u-sub. We can use the integral test since f(n)=an is positive, continuous, and decreasing. It’s a bit tricky to see at first, but we can use u=ln(ln n)) to rearrange the sum into an integral. The integral will converge if p>1 since u=ln(ln n)) will remain in the denominator.
Flipped Math
Flipped Math is an awesome resource with full packets, test prep, and corrective assignments for each lesson!
Math Libretexts
Math LibreTexts is a free online resource packed with textbooks and learning materials for math students. It simplifies complex concepts, provides practice examples, and offers interactive features for convenient learning anytime, anywhere. It’s like a superhero ally for math students, making math easier to understand and excel in.
Paul's Online Notes
Paul’s Online Notes is another amazing resource with in-depth explanations and a few examples. The second half of the article is about the p-series test.
Khan Academy
A classic resource for a reason, Khan Academy offers a ton of practice resources with explanatory videos. However, they’re known for being easier than the actual AP test so you might want to supplement with other resources to make sure you’re prepared for test day.