Sigma Notation Calculator

How to Use:

Enter a mathematical expression using 'i' as the index variable. The calculator will compute the sum of this expression for all integer values of i from the lower to upper bound.

Supported Syntax:

  • Basic operators: + - * / ( )
  • Exponents: Use ** or Math.pow(base, exp)
    Examples: i**2 or Math.pow(i, 2) for i²
  • Common functions: Math.sqrt(i), Math.abs(i), Math.floor(i), Math.ceil(i)
  • Constants: Math.PI, Math.E

Example Expressions:

  • i**2 → Sum of squares
  • 2*i + 1 → Sum of odd numbers
  • 1/i → Harmonic series
  • i*(i+1)/2 → Sum of triangular numbers
  • Math.sqrt(i) → Sum of square roots

Result: —

Calculation Steps:

What Sigma Notation Means

The sigma symbol is mathematical shorthand for "add up this expression for every integer value of the index, from a lower bound to an upper bound." The tool evaluates your expression once per integer in that range and adds the results.

Worked Example

The default expression i**2, summed from i=1 to i=5, represents the sum of i squared for i=1 to 5 - the sum of squares. Term by term: 1 squared=1, 2 squared=4, 3 squared=9, 4 squared=16, 5 squared=25. Adding them: 1+4+9+16+25 = 55. This is the exact calculation that runs automatically when the page loads.

A Second Example

The expression 2*i+1 summed from i=1 to i=6 gives terms 3, 5, 7, 9, 11, 13 (consecutive odd numbers, since 2i is always even and +1 makes it odd). Their sum: 3+5+7+9+11+13 = 48.

Writing Expressions

Use i as the index variable and standard operators (+, -, *, /). Exponents need either ** (i**2) or Math.pow(i, 2) - a single caret gets auto-converted to ** internally, so i^2 also works. Functions like Math.sqrt(i) and constants like Math.PI are available directly.

Range Limit

The tool caps the range at 1000 iterations (upper bound minus lower bound) to keep the browser responsive - a request like i=1 to i=1,000,000 would be rejected rather than attempted.