How this calculator works
A series adds sequence terms. Arithmetic terms have a constant difference; geometric terms have a constant ratio. A finite series always has a mathematical sum, though extreme inputs can exceed this tool's numeric range. An infinite geometric series converges when the ratio magnitude is below one. The all-zero series is a separate convergent case.
Formula
Arithmetic: Sₙ = n[2a+(n-1)d]/2. Geometric: Sₙ = a(1-rⁿ)/(1-r) for r ≠ 1; Sₙ = na for r = 1. Infinite geometric: S = a/(1-r) for |r| < 1.
Worked example
The arithmetic series 2+4+6+8+10 has sum 30. The first three terms of 4+2+1+... sum to 7, while the infinite geometric sum is 4/(1-1/2) = 8.
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Frequently asked questions
Can I enter a ratio as a fraction?
Yes. For example 1/3 or -2/5 can be entered directly in the ratio field.
What happens at ratio one?
Every term equals the first term. The finite sum is n times that term. The infinite sum diverges unless every term is zero.
Does an alternating series always converge?
No. Alternating geometric terms converge only if their magnitudes shrink towards zero, which requires |r| < 1 for a non-zero first term.