A geometric series is a sum in which each term is found by multiplying the previous term by the same constant ratio. These series matter because they model repeated growth, repeated decay, and patterns that get smaller step by step. In calculus, geometric series are one of the simplest examples of an infinite sum that can have a finite value.
They help build intuition for convergence, limits, power series, finance, and probability.
If the first term is a and the common ratio is r, the terms are a, ar, ar^2, ar^3, and so on. A finite geometric sum adds only a fixed number of terms, while an infinite geometric sum adds terms forever. The infinite sum converges only when the terms shrink fast enough, which happens when |r| < 1.
For example, 10 + 5 + 2.5 + 1.25 + ... has total 20 because each term is half the previous term.
Key Facts
- A geometric sequence has terms a, ar, ar^2, ar^3, ... where a is the first term and r is the common ratio.
- Finite geometric sum: S_n = a(1 - r^n)/(1 - r), for r != 1.
- If r = 1, the finite sum is S_n = na because every term is the same.
- Infinite geometric sum: S = a/(1 - r), valid only when |r| < 1.
- Convergence condition: an infinite geometric series converges if -1 < r < 1 and diverges if |r| >= 1.
- Worked example: 8 + 4 + 2 + 1 + ... has a = 8 and r = 1/2, so S = 8/(1 - 1/2) = 16.
Vocabulary
- Geometric series
- A geometric series is a sum of terms in which each term is obtained by multiplying the previous term by a fixed common ratio.
- Common ratio
- The common ratio is the constant multiplier r between consecutive terms in a geometric sequence.
- Partial sum
- A partial sum is the sum of the first n terms of a series.
- Convergence
- Convergence means that the partial sums of an infinite series approach a single finite number.
- Divergence
- Divergence means that the partial sums of an infinite series do not approach a finite limit.
Common Mistakes to Avoid
- Using S = a/(1 - r) when |r| >= 1. This is wrong because the infinite geometric sum formula only applies when the terms shrink toward zero.
- Confusing the first term a with the common ratio r. The first term is the starting value, while the ratio is the multiplier from one term to the next.
- Forgetting parentheses in S_n = a(1 - r^n)/(1 - r). Without the correct grouping, the order of operations can give a completely different answer.
- Using n instead of n - 1 in the last term. The nth term is ar^(n - 1), while ar^n is the term after the nth term.
Practice Questions
- 1 Find the sum of the first 6 terms of the geometric series 3 + 6 + 12 + 24 + ... .
- 2 Find the infinite sum of 12 + 4 + 4/3 + 4/9 + ... if it converges.
- 3 A geometric series has first term a = 5 and common ratio r = -1.2. Explain whether the infinite series converges or diverges, and justify your answer using the convergence condition.