What is sigma notation?

Sigma notation (using the Greek capital letter sigma) is a compact way to write the sum of many terms that follow a pattern. It specifies a formula in a variable i, a lower bound where i starts, and an upper bound where i finishes. For example, sigma from i equals 1 to 5 of i squared means 1 squared plus 2 squared plus 3 squared plus 4 squared plus 5 squared, which equals 55.

What are the formulas for common summations?

Three classic summation formulas are: the sum of i from 1 to n equals n times (n plus 1) divided by 2; the sum of i squared from 1 to n equals n times (n plus 1) times (2n plus 1) divided by 6; and the sum of i cubed from 1 to n equals the square of n times (n plus 1) divided by 2. These are known as the power sum formulas and are useful in calculus and discrete mathematics.

Can I use the summation calculator with a custom expression?

Yes. Select the custom option and type any mathematical expression in terms of i, using standard operators: i*2 for 2i, i^2 for i squared, sqrt(i) for the square root of i, 1/i for a harmonic sum, and so on. The calculator evaluates the expression for each integer value of i from the lower to the upper bound and sums the results.

Summation (Sigma) Calculator

Summation notation, written with the Greek capital letter sigma, is the standard way to express the sum of a series of terms that follow a mathematical pattern. The notation specifies three things: the formula that defines each term, the starting index (lower bound), and the ending index (upper bound). For example, the sum of i squared from i=1 to i=5 means you substitute each integer value of i from 1 to 5 into the formula i squared, then add all the results together: 1 + 4 + 9 + 16 + 25 = 55. This calculator handles four types of summation. The preset formulas, i, i squared, and i cubed, cover the three classical power sums that appear most often in calculus and discrete mathematics and have well-known closed-form answers. The custom option lets you enter any expression in terms of i, such as 2*i+1 for odd numbers, 1/i for a harmonic partial sum, or i^3-i for a more complex formula. You set the lower and upper bounds for i, and the calculator evaluates the expression at each integer step and returns the total, the running list of individual term values, the number of terms summed, and the mean term value. The lower bound can be zero or any positive integer; the upper bound must be at least equal to the lower bound. The custom expression parser uses standard mathematical notation with i as the variable, and supports the Math functions available in JavaScript such as sqrt, abs, and pow. Results are displayed to four decimal places for non-integer values.

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Formula

Lower bound (start of i)

Upper bound (end of i)

Custom expression (use i as variable)

total sum

Number of terms5

Mean term11.00

Max term25

i=1: 1, i=2: 4, i=3: 9, i=4: 16, i=5: 25

Terms shown for up to 100 steps. Custom expressions support i, +, -, *, /, ^, sqrt(), abs(), floor(), ceil(), and other Math functions.

How it works

For each preset formula, the calculator substitutes each integer value of i from the lower bound to the upper bound into the formula and accumulates the running sum. The preset formulas have known closed-form answers for verification: sum of i from 1 to n equals n(n+1)/2; sum of i squared from 1 to n equals n(n+1)(2n+1)/6; sum of i cubed from 1 to n equals [n(n+1)/2] squared. For custom expressions, the input string is evaluated using the JavaScript Function constructor with i as the variable, with Math functions available in scope via a with(Math) block. Invalid expressions return an error message rather than a numeric result.

Worked example

Using the defaults: formula = i squared, lower bound = 1, upper bound = 5. The individual terms are: i=1 gives 1 squared = 1; i=2 gives 4; i=3 gives 9; i=4 gives 16; i=5 gives 25. Total = 1 + 4 + 9 + 16 + 25 = 55. The closed-form check is 5 times 6 times 11 divided by 6 = 330 divided by 6 = 55, confirming the result. Mean term = 55 divided by 5 = 11.

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