Summation is Linear:

These properties may be used to expand terms and separate into different summations:

Summation is sequential – you can combine the indices in a natural way:

Reversing the indices has no effect – unlike as when the limits of integration are reversed:

and

These reflect that the order in which numbers are added is irrelevant to the total.

If the sign ofis not always constant, so that they are sometimes positive and sometimes negative then

Fallacies

You cannot separate the summation of a product into a product of summations:

It must always be remembered too that a complex series is often geometric, so thatis geometric with common ratioand the normal nth term and summation formulae can be used.