Infinite Geometric Series Formula

Infinite geometric series are the sum of all terms of a geometric sequence with common ratio |r| < 1.

The Formula

∑n=0∞arn=a1−rwhen ∣r∣<1\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \quad \text{when } |r| < 1
Partial sum: Sn=a⋅1−rn1−rS_n = a \cdot \frac{1 - r^n}{1 - r}. As n→∞n \to \infty, rn→0r^n \to 0 when ∣r∣<1|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

∑n=0∞12n=1+12+14+18+⋯=11−12=2\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

aa = first term, rr = common ratio. S∞=a1−rS_\infty = \frac{a}{1-r}.

What This Formula Means

The sum of all terms of a geometric sequence with common ratio ∣r∣<1|r| < 1. The infinite sum converges to a1−r\frac{a}{1-r}, where aa 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

∑n=0∞arn\sum_{n=0}^{\infty} ar^n converges iff ∣r∣<1|r| < 1, in which case ∑n=0∞arn=a1−r\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}. Proof: SN=a⋅1−rN+11−rS_N = a \cdot \frac{1-r^{N+1}}{1-r}, and ∣r∣<1  ⟹  rN+1→0|r| < 1 \implies r^{N+1} \to 0, so lim⁡N→∞SN=a1−r\lim_{N \to \infty} S_N = \frac{a}{1-r}.

Worked Examples

Example 1

easy
Find the sum of the infinite geometric series ∑n=0∞(13)n\displaystyle\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n.

Answer

32\frac{3}{2}

First step

1
Identify: first term a=1a = 1 (when n=0n=0), common ratio r=13r = \frac{1}{3}.

Full solution

  1. 2
    Check: ∣r∣=13<1|r| = \frac{1}{3} < 1, so the series converges.
  2. 3
    Sum formula: S=a1−r=11−13=123=32S = \frac{a}{1-r} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}.
Always verify ∣r∣<1|r| < 1 before applying the formula. The first term aa is the value at n=0n=0, which is (13)0=1\left(\frac{1}{3}\right)^0 = 1.

Example 2

medium
Convert the repeating decimal 0.27‾0.\overline{27} to a fraction using an infinite geometric series.

Example 3

medium
Find the sum of the infinite series 12+4+43+49+⋯12 + 4 + \frac{4}{3} + \frac{4}{9} + \cdots

Common Mistakes

  • Using a1−r\frac{a}{1-r} without checking ∣r∣<1|r|<1 - the formula only holds when the ratio shrinks the terms.
  • Plugging in the wrong aa - aa is the FIRST term of the sum, not the ratio or a later term.
  • Confusing the ratio with the difference - find rr by dividing consecutive terms, not subtracting them.

Why This Formula Matters

It is the first place students meet a finite answer for an infinitely long sum, which reshapes intuition before limits, repeating decimals, and Taylor series. The ∣r∣<1|r|<1 gate is the whole point: outside it the sum is meaningless, so the concept is really about WHEN summing forever is allowed. Recognizing it by "Are the terms a geometric sequence with ∣r∣<1|r|<1, and am I asked for the sum of all of them?" — rather than by familiar numbers — is what lets a student tell it apart from finite geometric series and infinite arithmetic series and divergent geometric series in a mixed problem set.

Frequently Asked Questions

What is the Infinite Geometric Series formula?

The sum of all terms of a geometric sequence with common ratio ∣r∣<1|r| < 1. The infinite sum converges to a1−r\frac{a}{1-r}, where aa 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?

aa = first term, rr = common ratio. S∞=a1−rS_\infty = \frac{a}{1-r}.

Why is the Infinite Geometric Series formula important in Math?

It is the first place students meet a finite answer for an infinitely long sum, which reshapes intuition before limits, repeating decimals, and Taylor series. The ∣r∣<1|r|<1 gate is the whole point: outside it the sum is meaningless, so the concept is really about WHEN summing forever is allowed. Recognizing it by "Are the terms a geometric sequence with ∣r∣<1|r|<1, and am I asked for the sum of all of them?" — rather than by familiar numbers — is what lets a student tell it apart from finite geometric series and infinite arithmetic series and divergent geometric series in a mixed problem set.

What do students get wrong about Infinite Geometric Series?

The procedure for infinite geometric series is the easy part; the trap is using a1−r\frac{a}{1-r} without checking ∣r∣<1|r|<1. Asking "Are the terms a geometric sequence with ∣r∣<1|r|<1, and am I asked for the sum of all of them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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