Infinite Geometric Series Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Infinite Geometric Series.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

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.

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.

Sense of Study hint: Identify the first term a and the common ratio r separately, verify |r| < 1, then plug into 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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the sum: 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + \cdots

Example 2

hard
For what values of x does \displaystyle\sum_{n=0}^{\infty} (x-1)^n converge, and what is the sum?
← Back to Infinite Geometric Series Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

geometric sequenceserieslimit