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Summation (Σ) Calculator

Math Calculators

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
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

if(i)Partial sum
111
223
336
4410
5515
6621
7728
8836
9945
101055
111166
121278
… 85 middle terms omitted …
98984,851
99994,950
1001005,050

Calculation steps

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

Common closed-form sums

SumClosed formNote
Σ i (1..n)n(n+1)/2first 100 = 5050
Σ i² (1..n)n(n+1)(2n+1)/6first 10 = 385
Σ i³ (1..n)[n(n+1)/2]²equals (Σi)²
Σ (2i-1) (1..n)n²sum of first n odds
Σ 1/2^i (0..n)2 − 1/2ⁿconverges to 2
Σ 1/(i(i+1)) (1..n)1 − 1/(n+1)telescoping, → 1
Σ 1/i (1..n)≈ ln(n) + γharmonic 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

  1. Enter values or expressions: type directly into the Summation (Σ) Calculator — keyboard friendly.
  2. Instant results: everything computes live as you type, no button needed.
  3. Follow the steps: the result panel shows full working and formula substitution.
  4. 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.

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Source & Updates

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

Last updated:

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

Maintained and periodically reviewed by the YiJianSuan editorial team — review process in ourEditorial Policy.

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.