Series Calculator
Free series calculator. Sum arithmetic series, geometric series (finite or infinite), custom lists of terms, and sigma notation with n. Shows formulas and steps.
Full original guide (expanded)
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
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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.
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Last code update: 2026-01-19
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