i.e. is the sum of all the numbers . A sum of one number is defined to be that number, so that

The ``'' in equation (3.53) is a dummy variable, and can be replaced by any symbol that has no meaning assigned to it. Thus

The following properties of the summation notation should be clear from the definition. (Here , and are integers with and and are real numbers.)

If for all satisfying then

Also

Using the summation notation, we can rewrite equations (2.9)
and (2.23) as

and

The use of the Greek letter to denote sums was introduced
by Euler in 1755[15, page 61].
Euler writes

Compare this with the notation in Bernoulli's table 2.2. (The apparent difference is due to the fact that for Euler, denotes the sum of squares, starting with , whereas for Bernoulli denotes the sum of squares starting with .) The use of the symbol (which is a form of ) for sums was introduced by Leibniz. The use of limits on sums was introduced by Augustin Cauchy(1789-1857). Cauchy used the notation to denote what we would write as [15, page 61].

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