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Sequences and series

Infinite geometric series (sum to infinity)

The total of a never-ending geometric series whose terms shrink, such as 1 + ½ + ¼ + … = 2.

S=a11−r, |r|<1

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024681012141618200.511.52S = 2
  • Partial sums: 1, 1.5, 1.75, 1.875, 1.9375, … → 2
  • After 20 terms the gap to the limit is 1.91 × 10^−6.

What the symbols mean

S
the sum to infinity, S∞
a1
the first term
r
the common ratio; it must be strictly between −1 and 1

When to use it

Use it when a geometric series goes on forever and the ratio is between −1 and 1: total distance of a ball that keeps bouncing, a recurring decimal such as 0.999… or 0.272727…, the area of a fractal. If |r|≥1 the terms do not shrink and there is no finite sum: the series diverges.

Worked example

Find 1 + 0.5 + 0.25 + 0.125 + …, which has a₁ = 1 and r = 0.5.

  1. Start from the formula.

    S=a11−r

  2. Substitute a1 = 1, r = 0.5 and work it out.

    S=11−0.5
    =10.5
    =2

Answer S = 2

Why it works

In Sn=a1(1−rn)/(1−r), the term rn gets closer and closer to 0 as n grows when |r|<1, so the partial sums approach a1/(1−r).

Common mistakes

  • Using it when |r|≥1. The sum 1 + 2 + 4 + … has no finite total, even though 1/(1−2)=−1.
  • Taking a₁ as the second term when the series starts part-way, for example after the first bounce.
  • Treating 0.999… as "slightly less than 1". With a₁ = 0.9 and r = 0.1 the sum is exactly 1.