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 of
is 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 that
is geometric with common ratio
and the normal nth term and summation formulae can be used.