Infinite Geometric Series Formula

The 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 to use: 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).

Quick 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

Notation

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

What This Formula Means

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.

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).

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}.

Worked Examples

Example 1

easy
Find the sum of the infinite geometric series \displaystyle\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n.

Solution

  1. 1
    Identify: first term a = 1 (when n=0), common ratio r = \frac{1}{3}.
  2. 2
    Check: |r| = \frac{1}{3} < 1, so the series converges.
  3. 3
    Sum formula: S = \frac{a}{1-r} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}.

Answer

\frac{3}{2}
Always verify |r| < 1 before applying the formula. The first term a is the value at n=0, which is \left(\frac{1}{3}\right)^0 = 1.

Example 2

medium
Convert the repeating decimal 0.\overline{27} to a fraction using an infinite geometric series.

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.

Why This Formula 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.

Frequently Asked Questions

What is the Infinite Geometric Series formula?

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.

How do you use the Infinite Geometric Series formula?

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).

What do the symbols mean in the Infinite Geometric Series formula?

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

Why is the Infinite Geometric Series formula important in Math?

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.

What do students get wrong about Infinite Geometric Series?

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.

What should I learn before the Infinite Geometric Series formula?

Before studying the Infinite Geometric Series formula, you should understand: geometric sequence, series, limit.

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