Geometric series: a + ar + ar² + ar³ + ... The nth term is aₙ = ar^(n-1). Partial sum: Sₙ = a(1-rⁿ)/(1-r). If |r|<1, the infinite sum converges: S∞ = a/(1-r). If |r|≥1, the series diverges. Common ratio r = aₙ₊₁/aₙ.
How to Use
Enter series parameters:
- First term (a): Starting value
- Ratio (r): Common ratio
- Terms (n): Number of terms
Convergence
- |r| < 1: converges to a/(1−r)
- |r| = 1: a+a+a+... diverges (unless a=0)
- |r| > 1: diverges (terms grow)
Applications
Compound interest, radioactive decay, fractal geometry (Koch snowflake), bouncing ball heights, mortgage payments, population models.
Step-by-Step Instructions
- 1Enter first term a.
- 2Enter common ratio r.
- 3Set number of terms n.
- 4View partial and infinite sums.
- 5Check convergence.