What is Summation (Sigma) Notation?
Sigma notation (named after the Greek capital letter sigma, Σ) is a shorthand way to write the sum of a sequence of numbers. Instead of writing out each term, you write a general formula and specify the range of values to substitute. The notation has three parts: the general term expression (what you are adding), the lower bound (where the index starts), and the upper bound (where the index stops).
For example, the notation "sum from n=1 to 5 of n squared" means: substitute n=1, n=2, n=3, n=4, and n=5 into n^2, then add the results: 1 + 4 + 9 + 16 + 25 = 55.
Common Summation Formulas
| Series | Expression | Closed-Form Sum (1 to n) | Example: n=10 |
|---|
| Natural numbers | Σ i | n(n+1)/2 | 55 |
| Squares | Σ i^2 | n(n+1)(2n+1)/6 | 385 |
| Cubes | Σ i^3 | [n(n+1)/2]^2 | 3,025 |
| Odd numbers | Σ (2i-1) | n^2 | 100 |
| Geometric (ratio r) | Σ r^i (i=0 to n) | (r^(n+1) - 1)/(r - 1) | Σ 2^i (0-9) = 1,023 |
| Constant | Σ c | c * n | Sum of 1, 1 to 10 = 10 |
How the Calculator Works
This calculator uses direct term-by-term evaluation. For each integer value of n from the start index to the end index, it substitutes n into your expression, evaluates the result, and adds it to a running total. This approach works for any expression that can be evaluated numerically, including polynomials, exponentials, fractions, and alternating series.
Supported operators and functions in the expression field: addition (+), subtraction (-), multiplication (*), division (/), powers (^), parentheses, and negative numbers. The index variable must be written as n (lower case). For example: n^2 + 3*n - 1, 2^n, 1/(n*(n+1)).
Arithmetic vs Geometric Series
An arithmetic series is one where each term increases by a constant difference. For example, 3 + 5 + 7 + 9 + 11 (difference of 2). The general term is f(n) = a + (n-1)*d, where a is the first term and d is the common difference. The sum of n terms is n*(first + last)/2.
A geometric series is one where each term is multiplied by a constant ratio. For example, 2 + 4 + 8 + 16 + 32 (ratio of 2). The general term is f(n) = a * r^(n-1). The sum of n terms is a*(r^n - 1)/(r - 1) for r not equal to 1. Both types are handled correctly by this calculator.
Practical Uses of Summation
Summation appears throughout mathematics and everyday life. In statistics, the mean of a dataset is the sum of all values divided by the count. In finance, compound interest involves sums of geometric terms. In physics, work done by a variable force is approximated by summing small increments. In computer science, the running time of algorithms is often expressed as a sum. Learning to read sigma notation and evaluate series is a key skill in mathematics from secondary school through to university level.
Related Calculators
Method: Direct term-by-term summation. The calculator substitutes each integer index value from the lower bound to the upper bound into the general term expression and accumulates the total. This approach is exact for any finite series with a mathematically valid expression.
This calculator evaluates finite sums only (lower and upper bounds must both be integers, with at most 10,000 terms). Expressions must use n as the index variable and standard mathematical operators. Infinite series convergence is not assessed. Results are displayed to sufficient decimal places for the values involved.
