BYTETOOLS

Summation Calculator

Evaluate a sigma-notation sum term by term with exact integer arithmetic, and see the standard closed-form identity beside the numeric answer.

385
Sum (exact)
10
Terms added
Exact integers
Arithmetic

Result

Σ (i = 1 to 10) of i^2 = 385

Every term is a whole number, so this total was computed with exact big-integer arithmetic — no rounding at all.

Closed form recognised — Sum of squares

Σ i² from 1 to 10 = n(n+1)(2n+1)/6 = 385

The algebraic identity gives 385, which matches the term-by-term total exactly.

Terms

Each index and the value of the term at that index
iterm
11
24
39
416
525
636
749
864
981
10100

What is the Summation Calculator?

The ByteTools Summation Calculator evaluates Σ from i = a to b of any expression in i. When every term is a whole number it sums with exact big-integer arithmetic, so the total is correct to the last digit no matter how large it grows — no floating-point rounding creeping in at the end.

  • Exact big-integer arithmetic whenever every term is a whole number
  • Recognises the Σi, Σi², Σi³, constant and geometric closed forms
  • Shows the algebraic identity beside the term-by-term total
  • Term table so you can see exactly what is being added
  • Handles the empty sum and the ratio-of-one geometric case safely
  • Runs entirely in your browser

How to use the Summation Calculator

  1. 1

    Type the expression to sum using i as the index, such as i^2 or 2^i.

  2. 2

    Set the lower limit a and the upper limit b.

  3. 3

    Read the total, and check whether it was computed with exact integers or floating point.

  4. 4

    Look at the closed-form panel if a standard identity was recognised.

  5. 5

    Click Copy result to save the sum and its formula.

About the Summation Calculator

The ByteTools Summation Calculator evaluates Σ from i = a to b of any expression in i. When every term is a whole number it sums with exact big-integer arithmetic, so the total is correct to the last digit no matter how large it grows — no floating-point rounding creeping in at the end.

It also recognises the standard closed forms: Σ i = n(n+1)/2, Σ i² = n(n+1)(2n+1)/6, Σ i³ = [n(n+1)/2]², the constant sum and the geometric series. When your expression matches one, the algebraic identity is shown beside the term-by-term total so you can check the formula against the arithmetic.

A table lists the individual terms so you can see what is being added. Everything runs in your browser — nothing is uploaded and it works offline.

Frequently asked questions

What does sigma notation mean?

The Σ symbol means 'add these up'. The index below it, such as i = 1, is where the counting starts; the number above is where it stops. The expression after Σ is evaluated at each index and the results are totalled.

What is the formula for the sum of the first n natural numbers?

It is n(n+1)/2. For example the numbers 1 to 100 add up to 100 × 101 ÷ 2 = 5,050. This calculator recognises that pattern and shows the identity alongside the computed total.

What happens if the upper limit is smaller than the lower limit?

That is the empty sum, which equals zero by convention because there are no terms to add. The calculator returns 0 and explains why rather than treating it as an error.

Why does my sum say it used floating point instead of exact integers?

Some term produced a value that is not a whole number — a division that does not divide evenly, a square root, or a trigonometric function. Those cannot be represented exactly, so the total carries a tiny rounding error.

What is the formula for a geometric series?

The sum of a geometric series is first × (1 − rⁿ) ÷ (1 − r), where r is the common ratio and n the number of terms. That formula divides by zero when r = 1, so this calculator handles that case separately by multiplying the constant term by the term count.

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