summation (sum)

A summation, also called a sum, is the result of arithmetically adding numbers or quantities. A summation always contains a whole number of terms. There can be as few as two terms, or as many as a hundred, a thousand, or a million. Some summations contain infinitely many terms.

For short sums, the numbers, or addends, can be written one after another, separated by addition signs (+). An example is 1/1 + 1/2 + 1/3. This becomes awkward when the number of terms is large. When a summation has infinitely many terms, the terms must occur in a definable sequence. This sequence is not always clear when the terms are simply listed and separated by + symbols. For this reason, the summation symbol was devised: a large, uppercase Greek letter sigma.

To denote 1/1 + 1/2 + 1/3, the following symbology can be used:

The expression on the left-hand side of the equation is the summation from n = 1 to n = 3 for 1/ n . The value of n is always an integer. It usually starts at 1 and always increases by 1 for each succeeding term in the summation.

Suppose the above series is extended without limit according to the obvious pattern. Then the summation from n = 1 to unlimited values of n for 1/ n , that is, 1/1 + 1/2 + 1/3 + 1/4 + ..., is denoted like this:

The sideways 8 means that n continues to increase without limit. It is imprecise to say we are dealing with the summation from n to infinity, but this terminology is often used anyway.

See our list of Mathematical Symbols .