1,187 Pages

Summation is the operation of adding a sequence of numbers to get a sum or total. It is usually denoted with the letter sigma. A sum of all the integers from 1 to 5 can be written as:

Any operation can be performed on . For instance,

A partial sum, where the sum is only of part of a series, is also called a finite sum. If a sum is between a number and infinity, it is called a series and is denoted

Infinite sums can be divergent, meaning they do not converge (such as or ), or convergent, meaning they equal a specific value (for instance,).

## Properties

A convergent sum of any series in which is multiplied by a constant is the same as the entire sum multiplied or divided by said constant. If is a constant then

Two convergent series added together with the same index are equal to the series sum of the arguments:

Some example sums with closed forms are shown below:

If a sum is geometric, or in the form

If the sum of a geometric series is infinite and convergent , the formula simplifies to:

If , then: