Index of summation - Sequences and Series

Here now, in

the i is termed as the index of summation or just index for short and note that the letter we employ to represent the index does not matter.  Thus for instance the following series are all the same.  The only dissimilarity is the letter we've utilized for the index.  

It is significant to again note that the index will start at doesn't matter whatever value the sequence of series terms starts or initiates at and this can literally be anything. Till now we've used n =0 and n = 1 but the index could have started anywhere.  Actually, we will generally use n a ∑ to denote an infinite series where the starting point for the index is not significant.  While we drop the initial value of the index we'll as well drop the infinity from the top so remember that it is still technically there. 

In these facts or theorems the starting point of the series will not influence the result and thus to simplify the notation and to prevent giving the impression that the starting point is significant we will drop the index from the notation.  Though do not forget, that there is a starting point and that this will be an infinite series.

Note though, that if we do put an initial value of the index on a series in a fact or theorem it is there as it really does need to be there. Now that few notational issues are out of the way we need to start thinking about several ways which we can manipulate series.