Sigma Calculator – Summation Notation Calculator
Use this free Sigma calculator to evaluate finite summations written in mathematical sigma notation. Enter an expression such as k², 2k + 1, 1/k, or 2^k, choose the starting and ending values, and calculate the total with a step-by-step expansion.
Calculate a Sigma Sum
What Is Sigma Notation?
Sigma notation is a compact way to represent the sum of many mathematical terms. It uses the uppercase Greek letter Σ, called sigma, to indicate summation.
A typical finite summation looks like this:
The variable k takes each value from the lower limit, 1, through the upper limit, n. The expression f(k) is evaluated for each value, and all of the resulting values are added together.
How Sigma Summation Works
Consider:
The variable k takes the values 1, 2, 3, 4, and 5. Therefore, the sigma expression expands to:
This becomes:
The calculator performs this expansion automatically.
Parts of a Sigma Expression
Σ
The Greek capital letter sigma indicates that values should be added together.
Lower limit
The value below sigma specifies where the summation starts.
Upper limit
The value above sigma specifies the final value included in the summation.
Expression
The expression to the right of sigma is evaluated for every value of the summation variable.
Sigma Notation Examples
Sum of the first 10 integers
This is:
Sum of squares
This gives:
Linear expression
The terms are:
Reciprocal sum
The calculator evaluates:
Sigma Calculator vs. Series Sum Calculator
Although sigma notation and series are closely related, these calculators serve different purposes.
Sigma Calculator
Use the Sigma Calculator when your problem is written as a general mathematical summation such as:
You specify the expression and the summation limits.
Sum of a Series Calculator
Use the Sum of a Series Calculator when you have an arithmetic or geometric sequence and want to calculate its sum.
For example, you can enter the first term, common difference or ratio, and number of terms.
Common Summation Formulas
Many frequently encountered sigma sums have closed-form formulas.
The Sigma Calculator evaluates the terms directly, which is useful when the expression does not have a convenient closed-form formula.
Related Summation and Math Calculators
Continue with another Summe.org calculator:
Frequently Asked Questions
What does Σ mean in mathematics?
The symbol Σ means summation. It instructs you to add the values of an expression over a specified range of the summation variable.
How do I calculate a sigma sum?
Substitute each value of the summation variable into the expression, calculate each resulting term, and add all of the terms together. This calculator performs those steps automatically.
What is the difference between sigma notation and a series?
Sigma notation is a compact way of writing a sum, while a series is the sum of the terms of a sequence. Sigma notation can be used to represent many types of finite sums.
Can I use negative starting values?
Yes. For example, you can calculate a sum from k = −5 through k = 5.
Can the summation use a step other than 1?
Yes. Enter a different integer step. For example, a step of 2 calculates values such as 1, 3, 5, 7, and so on.
Does this calculator support powers?
Yes. You can enter expressions such as k², k³, or 2^k.
More Summation Tools
If you already know that your sequence is arithmetic or geometric, use the Sum of a Series Calculator . For general sigma notation, use the calculator on this page.
You can also explore all available tools from the Summe.org mathematics calculator collection .