10.2 Working with Geometric Series

What is a geometric series?

A geometric series has the form Σ a·r^n, where each term is the previous term times a constant ratio r. Starting at n=0, the terms are a, ar, ar², ar³, .... Recognizing this pattern is key to quickly determining convergence and sum.

Convergence and sum formula

A geometric series converges if and only if |r| < 1. When it converges, its sum is S = a / (1 - r), where a is the first term. If |r| ≥ 1, the series diverges because terms don't shrink to zero, so partial sums don't approach a finite limit.

Manipulating geometric series

Many series can be rewritten to match the geometric form by factoring out constants or adjusting exponents. Recognizing hidden geometric series (e.g., rewriting 3^(n+1)/2^n) lets you apply the sum formula directly instead of using more complex tests, making them fast, reliable building blocks in series analysis.

Key terms

Geometric series form

Σ a·r^n, where a is the first term and r is the common ratio between consecutive terms.

Convergence condition

A geometric series converges if and only if |r| < 1.

Sum formula for convergent geometric series

S = a / (1 - r), where a is the first term and r is the common ratio.

Divergence condition

If |r| ≥ 1, the series diverges because the terms don't approach zero.

Common ratio (r)

The fixed multiplier between consecutive terms in a geometric series, found by dividing any term by the previous term.

Try an AP-style question

Σₙ₌₀^∞ 3(1/2)ⁿ = ?

1. Diverges
2. 3
3. 3/2
4. 6