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Semiformal Definition of a "Series":
A series an is the indicated sum of all values of an when n is set to each integer from a to b inclusive; namely, the indicated sum of the values aa + AA+1 + AA+2 + ... + ab-1 + ab.

Definition of the "Sum of the Series":
The "sum of the series" is the actual result when all the terms of the series are summed.

Note the difference: "1 + 2 + 3" is an example of a "series," but "6" is the actual "sum of the series."

Algebraic Definition: an = AA + AA+1 + AA+2 + ... + AB-1 + AB

Summation Arithmetic: c an = c an (constant c) an + bn = an + bn an - bn = an - bn

Summation Identities on the Bounds:
 b an n=a c + an   n=b+1 c = an   n = a

 b an n=a b-c = an+c    n=a-c
 b an n=a b/c = anc    n=a/c
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(similar relations exist for subtraction and division as generalized below for any operation g)
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 b an n=a g(b) = ag -1(c)    n=g(a) 