Library / Foundations: Maths / FMA-18
Foundations: Maths
Geometric series: sum to infinity
Total of a series whose terms shrink by a constant ratio.
- Ref
- FMA-18
- Rev
- v1
- Field
- Foundations: Maths
- Result
- S∞
a is the first term and r the common ratio. The sum to infinity exists only for |r| < 1. The finite sum is S_n = a(1 − rⁿ)/(1 − r).
When each term is a fixed fraction of the one before, the terms get so small that the total settles at a limit.
Calculator
- Sum of the first 10 terms7.992
- Remaining after 10 terms0.007813
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Ratios near 1 converge slowly; ratios near 0 converge after a few terms.
- Negative ratios give terms that alternate in sign and still converge.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Common ratio | −0.9 to 0.9 | 6.316 | 120 | 113.7 | +undefined% | — |
| a First term | 1 to 100 | 0.6667 | 66.67 | 66 | +10% | +10 % |
Sweep of Common ratio
| Common ratio | Sum to infinity |
|---|---|
| −0.9 | 6.316 |
| −0.675 | 7.164 |
| −0.45 | 8.276 |
| −0.225 | 9.796 |
| 0 | 12 |
| 0.225 | 15.48 |
| 0.45 | 21.82 |
| 0.675 | 36.92 |
| 0.9 | 120 |
Sweep one variable
Variables and how each one contributes
Sum to infinity
What the equation gives you. Everything else feeds it.
First term
The first term. Directly proportional.
Common ratio
Ratio of each term to the previous (|r| < 1). Strong near 1: as r approaches 1 the sum grows without limit.
Worked example with the values above
- Check convergence: |r| = 0.5 is below 1.
- Denominator: 1 − r = 1 − (−0.5) = 1.5.
- S∞ = 12 / 1.5 = 8.
Assumptions and limits
- The ratio is constant and |r| < 1.
- The series is continued indefinitely.
Common mistakes
- Using the formula when |r| ≥ 1.
- Forgetting a negative ratio changes the denominator.
Statistics and uncertainty
The sum is very sensitive to r near 1, since 1 − r is small: a 1% error in r = 0.95 gives about a 19% error in the sum.
Where it is used
- Decay and damping sequences such as repeated rebounds.
- Total effect of repeated diminishing contributions.
Maths you need
- Ratios: multiplying each term by r
- Fractions: dividing by 1 − r