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Basic Summation Operations - Manipulating of domain and change of variable

  1. Jul 9, 2010 #1
    1. The problem statement, all variables and given/known data
    a) Show that [tex]$\sum_{S(j)} a_j + \sum_{R(j)} a_j = \sum_{S(j) and R(j)} a_j + \sum_{S(j) or R(j)} a_j$[/tex] is valid for an arbitrary infinite series, provided that 3 out of 4 sums exist.

    b) Show that [tex]$\sum_{R(j)} a_j = \sum_{R(c-j)} a_{c-j}[/tex] 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:
    [tex]$\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)[/tex]

    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.
     
  2. jcsd
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