Infinite Geometric Series

Q: When does an infinite geometric series converge?

An infinite geometric series converges only when the absolute value of the common ratio satisfies |r| < 1, meaning r is between −1 and 1 (exclusive). When this condition holds, the terms get progressively smaller and approach zero, allowing the partial sums to settle toward a finite limit. If |r| ≥ 1, the terms do not shrink, so the series diverges.

Q: What is the difference between a geometric sequence and an infinite geometric series?

A geometric sequence is an ordered list of numbers where each term is found by multiplying the previous term by a constant ratio. An infinite geometric series is what you get when you add up all the terms of that sequence. The sequence 3, 1.5, 0.75, … is a list; the series 3 + 1.5 + 0.75 + … represents a sum.

Q: How do you convert a repeating decimal to a fraction using an infinite geometric series?

A repeating decimal can be written as an infinite geometric series. For example, 0.333… = 3/10 + 3/100 + 3/1000 + …, which is a geometric series with a = 3/10 and r = 1/10. Applying the formula gives S = (3/10)/(1 − 1/10) = (3/10)/(9/10) = 3/9 = 1/3. This technique works for any repeating decimal.

Infinite Geometric Series

An infinite series that is geometric. An infinite geometric series converges if its common ratio r satisfies –1 < r < 1. Otherwise it diverges.

Infinite geometric series formulas: General Form a₁+a₁r+a₁r²+…, Sum=a₁/(1−r) if −1<r<1, with two examples showing convergent sums.

Key Formula

S = \sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}, \quad |r| < 1

Where:

$S$ = The sum of the infinite geometric series
$a$ = The first term of the series
$r$ = The common ratio between consecutive terms
$n$ = The index of summation, starting from 0

Worked Example

Problem: Find the sum of the infinite geometric series: 12 + 6 + 3 + 1.5 + …

Step 1: Identify the first term a.

a = 12

Step 2: Find the common ratio r by dividing the second term by the first term.

r = \frac{6}{12} = \frac{1}{2}

Step 3: Check that the series converges by verifying |r| < 1.

\left|\frac{1}{2}\right| = \frac{1}{2} < 1 \quad \checkmark

Step 4: Apply the sum formula.

S = \frac{a}{1 - r} = \frac{12}{1 - \frac{1}{2}} = \frac{12}{\frac{1}{2}} = 24

Answer: The sum of the infinite geometric series is 24.

Another Example

This example involves a negative common ratio, which produces an alternating series. It shows how to handle the subtraction of a negative number in the denominator of the formula.

Problem: Find the sum of the infinite geometric series: 27 − 9 + 3 − 1 + …

Step 1: Identify the first term a.

a = 27

Step 2: Find the common ratio r. Since the signs alternate, expect a negative ratio.

r = \frac{-9}{27} = -\frac{1}{3}

Step 3: Check for convergence. The absolute value of r must be less than 1.

|r| = \left|-\frac{1}{3}\right| = \frac{1}{3} < 1 \quad \checkmark

Step 4: Substitute into the sum formula. Be careful with the double negative in the denominator.

S = \frac{27}{1 - \left(-\frac{1}{3}\right)} = \frac{27}{1 + \frac{1}{3}} = \frac{27}{\frac{4}{3}} = 27 \times \frac{3}{4} = \frac{81}{4} = 20.25

Answer: The sum of the infinite geometric series is 81/4, or 20.25.

Frequently Asked Questions

When does an infinite geometric series converge?

An infinite geometric series converges only when the absolute value of the common ratio satisfies |r| < 1, meaning r is between −1 and 1 (exclusive). When this condition holds, the terms get progressively smaller and approach zero, allowing the partial sums to settle toward a finite limit. If |r| ≥ 1, the terms do not shrink, so the series diverges.

What is the difference between a geometric sequence and an infinite geometric series?

A geometric sequence is an ordered list of numbers where each term is found by multiplying the previous term by a constant ratio. An infinite geometric series is what you get when you add up all the terms of that sequence. The sequence 3, 1.5, 0.75, … is a list; the series 3 + 1.5 + 0.75 + … represents a sum.

How do you convert a repeating decimal to a fraction using an infinite geometric series?

A repeating decimal can be written as an infinite geometric series. For example, 0.333… = 3/10 + 3/100 + 3/1000 + …, which is a geometric series with a = 3/10 and r = 1/10. Applying the formula gives S = (3/10)/(1 − 1/10) = (3/10)/(9/10) = 3/9 = 1/3. This technique works for any repeating decimal.

Infinite Geometric Series vs. Finite Geometric Series

	Infinite Geometric Series	Finite Geometric Series
Number of terms	Infinitely many terms	A specific number n of terms
Formula	S = a / (1 − r), requires \|r\| < 1	S_n = a(1 − rⁿ) / (1 − r), works for any r ≠ 1
Convergence condition	Only converges when \|r\| < 1	Always produces a finite sum (it has finitely many terms)
When to use	When summing all terms of a never-ending geometric pattern	When summing a known, fixed number of terms

Why It Matters

Infinite geometric series appear throughout precalculus and calculus as a foundational example of convergence. They provide a practical tool for converting repeating decimals to fractions and for solving real-world problems involving perpetual processes, such as calculating the total distance of a bouncing ball or the present value of a perpetuity in finance. Mastering this concept also prepares you for the study of power series and Taylor series in higher mathematics.

Common Mistakes

Mistake: Using the sum formula when |r| ≥ 1.

Correction: The formula S = a/(1 − r) is only valid when |r| < 1. If |r| ≥ 1, the series diverges and has no finite sum. Always check the convergence condition before applying the formula.

Mistake: Mishandling a negative common ratio in the denominator.

Correction: When r is negative, the denominator becomes 1 − (−|r|) = 1 + |r|. Students often write 1 − |r| instead, which gives the wrong answer. Substitute the actual signed value of r into the formula and simplify carefully.

Related Terms

Infinite Series — General category that includes geometric series
Geometric Series — Covers both finite and infinite cases
Convergent Series — A series whose partial sums approach a limit
Common Ratio — The constant multiplier r between terms
Diverge — What happens when |r| ≥ 1
Geometric Sequence — The ordered list of terms before summing
Series — The general concept of summing a sequence
Finite — Contrast to the infinite nature of this series