Education
Series Calculator
Last updated: June 19, 2026
A series calculator is a mathematical utility that computes the summation, terms, and convergence properties of a sequence of numbers. The calculator supports arithmetic, geometric, and infinite series, applying algebraic and calculus rules to determine whether a given series converges or diverges. It calculates the partial sum for a specified range and computes the sum to infinity for convergent geometric series. Math students and researchers use this tool to analyze sequence behavior and calculate discrete sums.
Pick arithmetic or geometric, enter the first term, the common difference or ratio, and how many terms. The calculator returns the nth term, the sum of the first n terms, the closed-form formula used, and a preview of the early terms.
Quick Answer
Calculate the sum, terms, and convergence of arithmetic, geometric, and power series. Enter your formula, start value, and end value to find the results.
Sequence type
What to compute
e.g. 2
e.g. 3
A positive whole number. · e.g. 10
Arithmetic a_n
29
Arithmetic sequences add the same number each step; geometric sequences multiply by the same ratio each step. Both have closed formulas for the nth term and the sum of the first n terms.
Examples
Arithmetic a₁=2, d=3, n=10
a_10 = 29, S_10 = 155
Arithmetic a₁=1, d=1, n=100
a_100 = 100, S_100 = 5,050
Geometric a₁=3, r=2, n=5
a_5 = 48, S_5 = 93
Geometric a₁=1, r=0.5, n=10
a_10 ≈ 0.00195, S_10 ≈ 1.998
How it works
The calculator uses the standard closed-form formulas. They are exact for any n; no looping is needed.
Arithmetic · a_n = a₁ + (n − 1) × d · S_n = n × (a₁ + a_n) / 2
Geometric · a_n = a₁ × r^(n − 1) · S_n = a₁ × (1 − rⁿ) / (1 − r)
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Frequently asked questions
An arithmetic sequence adds the same number d each step: 2, 5, 8, 11, … (d = 3). A geometric sequence multiplies by the same number r each step: 3, 6, 12, 24, … (r = 2). The same calculator handles both.
a_n is the nth term of the sequence. For arithmetic, a_n = a₁ + (n − 1) × d. For geometric, a_n = a₁ × r^(n − 1). The calculator shows the formula with your actual values plugged in.
For arithmetic, S_n = n × (a₁ + a_n) / 2, the average of the first and last term times the count. For geometric with r ≠ 1, S_n = a₁ × (1 − rⁿ) / (1 − r). When r = 1, every term is a₁, so S_n is just n × a₁.
Not in this version. For |r| < 1, the infinite sum converges to a₁ / (1 − r). For |r| ≥ 1, it diverges. Choose a large n in this calculator to see the partial sum approaching the limit.
The preview shows the first 8 terms (or fewer if n < 8) for readability. The closed-form result is exact for any n; the preview is just a check.
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