Geometric Series
Calculator
Results
- nth term
- 4,374
- Sum of terms
- 6,560
- Infinite sum (|r| < 1)
- 0
Results
| nth term | 4,374 |
| Sum of terms | 6,560 |
| Infinite sum (|r| < 1) | 0 |
formula-map diagram
- nth term
- 4,374
- Sum of terms
- 6,560
- Infinite sum (|r| < 1)
- 0
Formula map
Formula
Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r)= 4374
Note
Simplified model: these are exact results of the stated idealised formulas, assuming perfect shapes and exact inputs. Real measurements carry error, and the ellipse perimeter is a Ramanujan approximation. Verify independently before relying on these figures.
More in Mathematics
See all →Frequently asked questions
What inputs does this calculator take?+
The first term (a), the common ratio (r) between consecutive terms, and the number of terms (n) to sum.
What is the sum formula?+
For a finite series, the sum is Sn = a × (1 - rⁿ)/(1 - r), valid when r is not equal to 1. Each term is the previous term multiplied by r.
What happens when r equals 1?+
When r = 1, every term is identical to a, so the formula's denominator would be zero. In that case the sum is simply n × a.
Can this calculate an infinite sum?+
An infinite geometric series only converges to a finite sum when |r| < 1, in which case the sum is a/(1-r). If |r| ≥ 1, the sum grows without bound and has no finite value.
Why does a small ratio like 0.5 still let the series add up to a fixed number even with infinite terms?+
Because each successive term shrinks geometrically, the added amounts get smaller and smaller fast enough that their total approaches a limit rather than growing forever — this is the core idea behind convergent series.