What this calculator does
A series is a sum of the terms of a sequence. This page handles three cases: arithmetic series (add the same amount each time), geometric series (multiply by the same ratio each time), and any formula a(n) you type, such as 1/n² or (−1)ⁿ⁺¹/n. Use it for homework, for loan and savings patterns, or to see whether an infinite sum settles down.
The formulas
Geometric, first n terms: Sₙ = a(1 − rⁿ) ÷ (1 − r), for r ≠ 1
Geometric, infinitely many terms: S = a ÷ (1 − r), only when −1 < r < 1
When you type whole numbers, decimals or fractions, these are computed in exact fractions, so 3 ÷ (1 − (−1/3)) shows as 9/4, not 2.2499999.
When does an infinite series converge?
A partial sum, however long, can't prove convergence on its own. So the calculator only gives a verdict when a standard test settles it for certain:
| Test | Series | Result |
|---|---|---|
| Geometric | Σ a·rⁿ | Converges if |r| < 1, diverges if |r| ≥ 1 |
| p-series | Σ 1/nᵖ | Converges if p > 1, diverges if p ≤ 1 |
| nth-term (divergence) test | Any Σ aₙ | If aₙ doesn't approach 0, the series diverges |
| Limit comparison with a p-series | Ratios of polynomials in n | Behaves like 1/nᵏ, where k is the degree gap |
| Alternating series test | (−1)ⁿ times a shrinking term | Converges if the size falls steadily to 0 |
The calculator recognizes terms built from rⁿ, powers of n and ratios of polynomials in n, including a (−1)ⁿ sign. For anything else, such as sin(n)/n², it adds the terms but says plainly that no test was applied.
Worked examples
- Arithmetic: 3 + 7 + 11 + … for 10 terms. The last term is 3 + 9 × 4 = 39, so S₁₀ = 10 × (3 + 39) ÷ 2 = 210.
- Geometric: 1 + 1/2 + 1/4 + … forever. Here a = 1 and r = 1/2, so S = 1 ÷ (1 − 1/2) = 2.
- p-series: Σ 1/n² from 1 to 100 is about 1.634984. Since p = 2 > 1 it converges, and the full sum is π²/6 ≈ 1.644934.
- Harmonic series: Σ 1/n is a p-series with p = 1, so it diverges, even though its terms shrink to 0. After 100 terms it's only about 5.1874, and it keeps growing without limit.
Limitations
- The partial sum adds up to 200,000 terms. For slowly converging series, the sum after N terms can still be far from the limit. For p-series with p > 1, the page shows bounds for the full sum from the integral test.
- Series of functions (power series, Taylor series) and tests like the ratio test on factorials aren't covered.
- Every term from the starting n onward must be defined. A term like 1/n starting at n = 0 is rejected.
Sources
- OpenStax Precalculus 2e, chapter 11 key equations (arithmetic and geometric series sums)
- OpenStax Calculus Volume 2, chapter 5 key concepts (geometric series, divergence test, p-series)
- OpenStax Calculus Volume 2, chapter 5 key equations (remainder estimate from the integral test)
- NIST Digital Library of Mathematical Functions, §25.6: ζ(2) = π²/6 and ζ(4) = π⁴/90