Hence, e.g., if ,

We write

where is a dummy index, and we think of as the largest of the numbers . By definition

and

Proof: Let be the proposition form on
such that is the
proposition (3.92). Then says

i.e.,

Hence is true.

Now for all ,

We also have

so

By induction, is true for all .

`a^n`

.

The notation for the factorial of was introduced by Christian Kramp in 1808[15, vol 2, p 66].

The use of the Greek letter to denote sums was introduced
by Euler
in 1755[15, vol 2,p 61].
Euler writes

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, vol 2, p 61].

In Maple,
the value of
is denoted
by `sum(f(i),i=1..n)`

. In Mathematica
it
is denoted by `Sum[f[i],{i,1,n}]`

.