Series Calculator
Compute the sum of arithmetic series, geometric series (finite or infinite), and custom series. You can also evaluate a sigma sum ∑ f(n) by entering the expression in n. Designed to cover the patterns found in Symbolab and Mathway examples, but in a clean, fast interface.
We use Sₙ = n/2 × [2a₁ + (n − 1)d] and aₙ = a₁ + (n − 1)d.
Series formulas used
Arithmetic series (n terms): \(S_n = \frac{n}{2} [2a_1 + (n - 1)d]\)
Arithmetic nth term: \(a_n = a_1 + (n - 1)d\)
Geometric series (n terms): \(S_n = a_1 \frac{1 - r^n}{1 - r}, \; r \neq 1\)
Infinite geometric series: \(S = \frac{a_1}{1 - r}\) if \(|r| < 1\)
Sigma: \(\sum_{n = m}^{k} f(n)\) = evaluate \(f(n)\) at every integer and add.
Notes & limitations
This is a practical series calculator focused on typical pre-calculus / calculus 2 tasks. For power series expansion, ratio/root tests, or symbolic summation of very complicated terms, you should still use a CAS like Symbolab or WolframAlpha.
However, for arithmetic/geometric/sigma sums, this tool is faster because it requires no special syntax and shows you the exact formula used.
FAQs
What if r = 1 in a geometric series?
Then every term is a₁, so Sₙ = n × a₁.
Why is infinite geometric not working for |r| ≥ 1?
Because such a series diverges — it does not have a finite sum.
How many terms can I put in custom list?
A few hundred is fine for quick browser calculations.
Formula (LaTeX) + variables + units
','\
Arithmetic series (n terms): \(S_n = \frac{n}{2} [2a_1 + (n - 1)d]\) Arithmetic nth term: \(a_n = a_1 + (n - 1)d\) Geometric series (n terms): \(S_n = a_1 \frac{1 - r^n}{1 - r}, \; r \neq 1\) Infinite geometric series: \(S = \frac{a_1}{1 - r}\) if \(|r| < 1\) Sigma: \(\sum_{n = m}^{k} f(n)\) = evaluate \(f(n)\) at every integer and add.
- T = property tax (annual or monthly depending on input) (currency)
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.