Infinite Geometric Series

Calculus
principle

Also known as: geometric series sum, sum to infinity, infinite-series

Grade 9-12

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The sum of all terms of a geometric sequence with common ratio |r| < 1. Geometric series appear everywhere: repeating decimals (0.333\ldots = \frac{1}{3}), compound interest, present value calculations in finance, probability, and as the simplest example of a convergent infinite series.

Definition

The sum of all terms of a geometric sequence with common ratio |r| < 1. The infinite sum converges to \frac{a}{1-r}, where a is the first term.

💡 Intuition

If each term is a fixed fraction of the previous one, the terms shrink fast enough that the total sum stays finite. Imagine walking halfway to a wall, then half the remaining distance, then half again—you approach the wall but the total distance is finite (exactly the full distance to the wall).

🎯 Core Idea

An infinite geometric series converges if and only if |r| < 1. The sum formula comes from the partial sum formula by taking the limit as n \to \infty: since |r| < 1, the r^n term vanishes.

Example

\sum_{n=0}^{\infty} \frac{1}{2^n} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{1}{1 - \frac{1}{2}} = 2

Formula

\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \quad \text{when } |r| < 1
Partial sum: S_n = a \cdot \frac{1 - r^n}{1 - r}. As n \to \infty, r^n \to 0 when |r| < 1.

Notation

a = first term, r = common ratio. S_\infty = \frac{a}{1-r}.

🌟 Why It Matters

Geometric series appear everywhere: repeating decimals (0.333\ldots = \frac{1}{3}), compound interest, present value calculations in finance, probability, and as the simplest example of a convergent infinite series. They are the gateway to understanding convergence.

💭 Hint When Stuck

Identify the first term a and the common ratio r separately, verify |r| < 1, then plug into a/(1-r).

Formal View

\sum_{n=0}^{\infty} ar^n converges iff |r| < 1, in which case \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}. Proof: S_N = a \cdot \frac{1-r^{N+1}}{1-r}, and |r| < 1 \implies r^{N+1} \to 0, so \lim_{N \to \infty} S_N = \frac{a}{1-r}.

🚧 Common Stuck Point

The formula \frac{a}{1-r} only works when |r| < 1. If |r| \geq 1, the series diverges (the terms don't shrink to zero). Always check the convergence condition first.

⚠️ Common Mistakes

  • Using the formula when |r| \geq 1: the series 1 + 2 + 4 + 8 + \ldots has r = 2, so it diverges—there is no finite sum.
  • Getting the first term a wrong: in \sum_{n=1}^{\infty} 3 \cdot (0.5)^n, the first term is a = 3(0.5) = 1.5, not a = 3. The formula uses the actual first term of the sum.
  • Confusing the partial sum formula S_n = a\frac{1-r^n}{1-r} with the infinite sum formula S_\infty = \frac{a}{1-r}—the partial sum still has the r^n term.

Frequently Asked Questions

What is Infinite Geometric Series in Math?

The sum of all terms of a geometric sequence with common ratio |r| < 1. The infinite sum converges to \frac{a}{1-r}, where a is the first term.

What is the Infinite Geometric Series formula?

\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \quad \text{when } |r| < 1
Partial sum: S_n = a \cdot \frac{1 - r^n}{1 - r}. As n \to \infty, r^n \to 0 when |r| < 1.

When do you use Infinite Geometric Series?

Identify the first term a and the common ratio r separately, verify |r| < 1, then plug into a/(1-r).

How Infinite Geometric Series Connects to Other Ideas

To understand infinite geometric series, you should first be comfortable with geometric sequence, series and limit. Once you have a solid grasp of infinite geometric series, you can move on to convergence divergence and taylor series.

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