A geometric series is a sum whose terms are made by multiplying by the same constant ratio each time. It appears in calculus whenever repeated scaling creates an infinite process, such as bouncing heights, compound interest models, fractals, and repeating decimals. The key question is whether the infinitely many terms add to a finite value or grow without bound.
This makes geometric series one of the first and most important examples of convergence.
Key Facts
- Geometric sequence terms have the form a, ar, ar^2, ar^3, ...
- A finite geometric series with n terms has sum S_n = a(1 - r^n)/(1 - r), for r != 1.
- An infinite geometric series converges only when |r| < 1.
- If |r| < 1, the infinite sum is S = a/(1 - r).
- If |r| >= 1, the infinite geometric series diverges and has no finite sum.
- A repeating decimal can often be written as a geometric series, such as 0.333... = 0.3 + 0.03 + 0.003 + ... = 1/3.
Vocabulary
- Geometric sequence
- A list of numbers in which each term is found by multiplying the previous term by a constant ratio.
- Geometric series
- The sum of the terms of a geometric sequence.
- Common ratio
- The constant multiplier r used to get from one term to the next in a geometric sequence.
- Convergence
- The behavior of an infinite series whose partial sums approach a finite number.
- Partial sum
- The sum of the first n terms of a series.
Common Mistakes to Avoid
- Using S = a/(1 - r) without checking |r| < 1 is wrong because the infinite sum formula only works for convergent geometric series.
- Confusing the first term a with the ratio r is wrong because a sets the starting size while r controls how terms change.
- Forgetting parentheses in S_n = a(1 - r^n)/(1 - r) is wrong because a small order-of-operations error can change the entire value.
- Treating every repeating decimal as starting at the tenths place is wrong because the first repeating block determines the first term and the ratio.
Practice Questions
- 1 Find the sum of the infinite geometric series 5 + 2.5 + 1.25 + 0.625 + ...
- 2 Find the fraction form of the repeating decimal 0.272727... by writing it as a geometric series.
- 3 A geometric series has first term 12 and common ratio -0.8. Explain whether it converges and describe how the signs and sizes of the terms behave.