Summation (Σ) Calculator

Sum any expression term by term · partial sums · closed forms

Quick Answer

Summation (Σ) Calculator: Sum any expression term by term · partial sums · closed forms. Summation (Σ) Calculator uses standard math formulas to compute accurate results instantly.

Syntax: i as index · supports + - * / ^ ( ) ! % · sqrt ln log sin cos abs pi e · implicit multiplication like 2i

Expression f(i)

Lower bound

Upper bound

Σ f(i)，i = 1 → 100

5,050

Terms100

First f(1)1

Last f(100)100

Average term50.5

Term-by-term & partial sums

i	f(i)	Partial sum
1	1	1
2	2	3
3	3	6
4	4	10
5	5	15
6	6	21
7	7	28
8	8	36
9	9	45
10	10	55
11	11	66
12	12	78
… 85 middle terms omitted …
98	98	4,851
99	99	4,950
100	100	5,050

Calculation steps

1Expression: f(i) = i, sum from i = 1 to 100, 100 terms in total
2First term f(1) = 1; last term f(100) = 100
3Evaluate every term and accumulate one by one
4Σ = 5,050; average term value = 50.5

Common closed-form sums

Sum	Closed form	Note
Σ i (1..n)	n(n+1)/2	first 100 = 5050
Σ i² (1..n)	n(n+1)(2n+1)/6	first 10 = 385
Σ i³ (1..n)	[n(n+1)/2]²	equals (Σi)²
Σ (2i-1) (1..n)	n²	sum of first n odds
Σ 1/2^i (0..n)	2 − 1/2ⁿ	converges to 2
Σ 1/(i(i+1)) (1..n)	1 − 1/(n+1)	telescoping, → 1
Σ 1/i (1..n)	≈ ln(n) + γ	harmonic diverges slowly
Σ i! (1..n)	无闭式	no closed form

This tool sums numerically term by term — it works for any expression even when no closed form exists. Trig functions use radians. Watch out: the harmonic series Σ 1/i diverges (grows like ln n), while Σ 1/2^i converges to 2. A finite numeric sum cannot prove convergence — it only approximates.

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How to Use

Enter values or expressions: type directly into the Summation (Σ) Calculator — keyboard friendly.
Instant results: everything computes live as you type, no button needed.
Follow the steps: the result panel shows full working and formula substitution.
Copy or recompute: one-click copy; change any input to recalculate instantly.

Key Points

Summation (Σ) Calculator is a free online math tool — open and use, no sign-up.
All calculations run locally in your browser; your data never leaves the device.
Key capability: Sum any expression term by term · partial sums · closed forms.
Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Summation (Σ) Calculator formula

Summation (Σ) Calculator uses standard math formulas to compute accurate results instantly.

FAQ

How do I use the Summation (Σ) Calculator?

Open the page and fill in the input fields — results update automatically. Sum any expression term by term · partial sums · closed forms.

Are the results accurate?

Yes. All calculations use standard mathematical formulas and are verified for accuracy.

Is it free? Do I need to sign up?

Completely free. No registration, login or download required — just open and use.

Does it work on mobile?

Yes. The responsive design works on phones, tablets and desktops.

Is my data saved?

No. All calculations run locally in your browser. Nothing is uploaded or stored.

Related Guides

Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated: 2026-08-07

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Cite this page: YiJianSuan. Summation (Σ) Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/sigma-sum

Results provided by this tool are for reference only and do not constitute professional advice.

Term-by-term & partial sums

Common closed-form sums

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How to Use

Key Points

Formula & How It Works

FAQ

Related Tools

Related Guides

Source & Updates

Term-by-term & partial sums

Common closed-form sums