an example showing the summation of squares: In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as , , , and ; the latter is also often used for the upper bound of a summation. Alternatively, index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to n. For example, one might write that: Generalizations of this notation are often used, in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example: is an alternative notation for the sum of over all (integers) in the specified range. Similarly, is the sum of over all elements in the set , and is the sum of over all positive integers dividing . There are also ways to generalize the use of many sigma signs. For example, is the same as A similar notation is used for the product of a sequence, where , an enlarged form of the Greek capital letter pi, is used instead of Special cases It is possible to sum fewer than 2 numbers: If the summation has one summand , then the evaluated sum is . If the summation has no summands, then the evaluated sum is zero, because zero is the identity for addition. This is known as the empty sum. These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if in the definition above, then there is only one term in the sum; if , then there is none. Formal definition Summation may be defined recursively as follows: , for b < a; , for b ≥ a. Measure theory notation In the notation of measure and integration theory, a sum can be expressed as a definite integral, where is the subset of the integers from to , and where is the counting measure. Calculus of finite differences Given a function that is defined over the integers in the interval , the following equation holds: This is the analogue of the fundamental theorem of calculus in calculus of finite differences, which states that: where is the derivative of . An example of application of