Math
Sum of Series Calculator
Add up the terms of an arithmetic series (constant difference) or a geometric series (constant ratio) using the closed-form formulas instead of adding term by term.
For geometric series you can also sum to infinity, and the calculator checks whether |r| < 1 so the sum actually converges.
For an infinite sum, |r| must be less than 1.
Try an example
Result
Sum
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- Last term
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- Formula used
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- Partial sums
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- Infinite sum
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
Sequence vs series
A sequence is a list of terms; a series is what you get when you add them. Sum formulas let you skip the adding: Gauss famously summed 1 + 2 + ... + 100 as 100 × 101 / 2 = 5,050 by pairing the first and last terms.
Arithmetic series
Pair the first term with the last, the second with the second-to-last, and so on. Every pair sums to a₁ + aₙ, and there are n/2 pairs. Arithmetic series never converge to a finite infinite sum because the terms do not shrink.
Geometric series
| Ratio r | Behaviour | Infinite sum |
|---|---|---|
| 0 < r < 1 | terms shrink, partial sums level off | a/(1 - r) |
| -1 < r < 0 | terms alternate and shrink | a/(1 - r) |
| r = 1 | constant terms | diverges |
| r > 1 or r < -1 | terms grow | diverges |
Worked examples
- 1 + 4 + 7 + ... (10 terms): a₁₀ = 28, S₁₀ = 10 × (1 + 28)/2 = 145.
- 1 + 1/2 + 1/4 + ... (10 terms): S₁₀ = (1 - 0.5^10)/(1 - 0.5) = 1.998046875; to infinity, S = 2.
- 3 + 6 + 12 + ... (8 terms): S₈ = 3(1 - 2^8)/(1 - 2) = 765; no infinite sum since r = 2.
How to use it
- Choose arithmetic (add the same amount each step), geometric (multiply by the same ratio), or infinite geometric.
- Enter the first term, then the difference or ratio, and the number of terms for a finite sum.
- Read the sum, the last term, the partial sums, and for geometric series whether the infinite sum exists.
How to read the answer
The sum is the total of all terms. Partial sums show the running total after 1, 2, 3, ... terms. A geometric series with |r| < 1 has partial sums that level off at a / (1 - r); with |r| ≥ 1 they run away and the infinite sum does not exist.
Common mistakes and edge cases
- Using the sequence formula (nth term) when the question asks for the series (sum).
- Summing an infinite geometric series with |r| ≥ 1; it diverges.
- Forgetting that the geometric sum formula fails for r = 1, where the sum is simply n × a.
- Counting n from 0 instead of 1 in the formulas.
Worked examples
Geometric, 10 terms
1 + 1/2 + 1/4 + ... to 10 terms
Sum
1.99804688
Infinite geometric
1 + 1/2 + 1/4 + ... forever = 2
Sum
2
Arithmetic
1 + 4 + 7 + ... (10 terms) = 145
Sum
145
Decreasing arithmetic
20 + 18 + 16 + ... (6 terms) = 90
Sum
90
Growing geometric
3 + 6 + 12 + ... (8 terms) = 765
Sum
765
Ratio of 1
5 + 5 + 5 + 5 = 20
Sum
20
Alternating infinite
1 - 1/3 + 1/9 - ... = 3/4
Sum
0.75
Divergent
Infinite sum with r = 2 does not exist
Sum
Error
Frequently asked questions
What is the formula for the sum of a series?+
Arithmetic: S_n = n(a₁ + aₙ)/2. Geometric: S_n = a(1 - rⁿ)/(1 - r). Infinite geometric with |r| < 1: S = a/(1 - r).
When does an infinite geometric series have a sum?+
Only when the common ratio satisfies |r| < 1, so the terms shrink toward zero. Then the sum is a/(1 - r).
What is 1 + 1/2 + 1/4 + 1/8 + ...?+
2. It is geometric with a = 1 and r = 1/2, so S = 1/(1 - 1/2) = 2.
Can an arithmetic series have an infinite sum?+
No (unless every term is 0). The terms do not shrink, so the partial sums grow without bound.
What is a partial sum?+
The sum of the first k terms, written S_k. The calculator lists S₁, S₂, S₃, ... so you can see the running total.
How did Gauss add 1 to 100 so fast?+
He paired 1 + 100, 2 + 99, ..., 50 + 51: fifty pairs of 101, giving 5,050. That is the arithmetic series formula n(a₁ + aₙ)/2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026