Basic Summation Operations - Manipulating of domain and change of variable

[PhDorBust]

Homework Statement

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.

Homework Equations

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

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.

 

[kurt.physics]

I am assuming there is some trick or insight that I am missing.UpdateAfter a great deal of thought I think I have found the answer to part b.$\sum_{R(j)} a_j = (\lim_{n\to\infty} \sum_{R(j),-n<j<0} a_j) + (\lim_{n\to\infty} \sum_{R(j), 0<=j<n} a_j)$$\sum_{R(c-j)} a_{c-j} = (\lim_{n\to\infty} \sum_{R(c-j),-n<c-j<0} a_{c-j}) + (\lim_{n\to\infty} \sum_{R(c-j), 0<=c-j<n} a_{c-j})$$= (\lim_{n\to\infty} \sum_{R(j),-(n+c)<j<-c} a_j) + (\lim_{n\to\infty} \sum_{R(j), c<=j<n+c} a_j)$$= (\lim_{n\to\infty} \sum_{R(j),-n<j<0} a_j) + (\lim_{n\to\infty} \sum_{R(j), 0<=j<n} a_j)$$= \sum_{R(j)} a_j$So $\sum_{R(j)} a_j = \sum_{R(c-j)} a_{c-j}$ is valid.