# Basic Summation Operations - Manipulating of domain and change of variable

1. Jul 9, 2010

### PhDorBust

1. The problem statement, all variables and given/known data
a) Show that $$\sum_{S(j)} a_j + \sum_{R(j)} a_j = \sum_{S(j) and R(j)} a_j + \sum_{S(j) or R(j)} a_j$$ is valid for an arbitrary infinite series, provided that 3 out of 4 sums exist.

b) Show that $$\sum_{R(j)} a_j = \sum_{R(c-j)} a_{c-j}$$ for an arbitrary infinite series where c is an integer.

2. Relevant equations
Since we are dealing with infinite series we cannot use the standard notation, we must use calculus so our definition of summation becomes:
$$\sum_{S(j)} a_j = (\lim_{n\to\infty} \sum_{S(j),-n<j<0} a_j) + (\lim_{n\to\infty} \sum_{S(j), 0<=j<n} a_j)$$

3. The attempt at a solution
I really haven't the slightest idea, I have been aimlessly trying to manipulate definition of infinite series with zero luck. These problems are from Knuth's TAOCP v.1.