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Summation Calculator

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Evaluate a summation (sigma notation) - enter an expression in terms of i, and the start and end values.

Summation Calculator

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Sum
385
Supports +, -, *, /, ^, parentheses, sin, cos, tan, sqrt, abs, and the variable i.

The formula

Sum = f(start) + f(start+1) + f(start+2) + ... + f(end)
Example

∑(i=1 to 10) of i²: 1+4+9+16+25+36+49+64+81+100 = 385.

Step-by-step guide

  1. Enter your expression in terms of i, like "i^2" or "2*i+1".
  2. Enter the start and end values for i.
  3. Read the sum.

What sigma notation compresses

The Greek letter sigma (∑) is mathematical shorthand for "add up all of these" - instead of writing out a long addition problem term by term, sigma notation specifies a pattern (the expression), a starting point, and a stopping point, letting you describe sums with hundreds or even infinitely many terms in a single compact line. It's foundational throughout statistics (like calculating a mean or variance), calculus (as the basis for integration), and computer science (analyzing how algorithms scale).

Common mistakes

Writing "2i" instead of "2*i" - like other calculators on this site, multiplication needs an explicit asterisk between a number and a variable.
Getting the start and end values backward - the end value must be greater than or equal to the start for the sum to make sense in this calculator.

Frequently asked questions

Can the expression include more than one variable?

This calculator supports a single summation variable, i - constants and functions of i (like i², sin(i), or 2*i+1) all work, but it doesn't support nested double summations.

Is there a limit to how large the range can be?

This calculator handles ranges up to 1,000,000 terms, evaluating each one directly - more than enough for the vast majority of coursework and practical calculations.

How is summation related to integration?

Integration can be thought of as the continuous version of summation - instead of adding discrete terms at whole-number steps, it "sums" infinitely many infinitesimally small slices, which is exactly the intuition behind the Riemann sum definition of an integral.

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