Given a function f(x), we know its even part is given as and its odd part is given by .
Consider a discrete sequence given by for , then the sum of the terms in the sequence till n terms is given as
Suppose I wanted to get the sum of the even terms in the above expression; then
and, for odd,
But wait, our sequence was defined for . Well, here's the thing: is some function of j, extending the domain to negative integers and evaluating the function gives the right answer... but I can't understand why the continuous function trick extended to here.
Sum of first n numbers given as , sum of first n odds will be given as:
My attempt at finding an exact connection: To the discrete sequence , we can associate a function and we can think of the summation as summing this function at several different input points i.e:
Then we apply the even odd decomposition and return back to the sequence world.
My question: Does there exist an association for every sequence with a function? If not, what is the criterion for an association to exist?