What is a geometric series?
On this episode of quick and easy AP Calc lessons: Geometric series! Although geometric series sound fancy, they’re actually one of the easiest series to work with (at least for the purpose of AP Calc)! The CED defines a geometric series as a series with a constant ratio between successive terms. In essence, it’s a series that has one initial term, a, and a ratio used to obtain the rest of the terms. NOTE: a is used as notation for the first term in the series.
See how a is being multiplied by a constant ratio for each successive term?
When does it diverge and converge?
A geometric series with ratio r diverges if |r | > 1 or r = 1.
If 0 < |r | < 1, then the series converges to the sum:
An important note here is that the ratio does not need to be positive. It converges to the above sum as long as the absolute value is between 0 and 1!
Partial Sum of Geometric Series
Although not in the CED, knowing the formula for partial sums of geometric series may be useful. Instead of manually adding up every term to get the nth partial sum, you can just use this formula.
Example 1
Given the infinite geometric series 
Find if the following converge or diverge.
If the series converges, find the infinite sum.
We can factor out a 2 and put the exponent around the ratio of 1/5 (We can do this since 1 raised to any power will always be 1). Since 0<|1/5|<1, the geometric series converges. We get the first term 2 and the common ratio 1/5, plug them into the formula for the infinite sum, and we have our answer!
Here we can just use approximations to recognize that the absolute value of our ratio is less than one, so the series converges. Since the sum starts at n=1, we plug n=1 in for our first term. We then use the formula to calculate the sum.
Using properties of exponents to manipulate the sum, we get r=5/3 which is larger than 1, so by the geometric series test the series diverges.
More Practice
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 first part of the article linked below explains geometric series!
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.