Hi Guys,

Can anybody tell me why it holds that:

I got it from my teacher, who left for vacation. I'm really confused by the change in indices.

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- September 13th 2010, 07:44 PMleoemilIndex change in Double summation
Hi Guys,

Can anybody tell me why it holds that:

I got it from my teacher, who left for vacation. I'm really confused by the change in indices. - September 14th 2010, 02:52 AMHallsofIvy
Look at some simple examples, with, say, T= 3. The outer sum then goes t= 1, t= 2, t= 3: for each such t, the term being added is

t= 1)

t= 2)

t= 3)

(Adding "1" from y= 1 to t just gives "t")

That entire sum is .

On the right side, again with T= 3, y takes on values of 1, 2, 3 and the inner sum there is

y= 1)

y= 2)

y= 3)

Multiplying each of those by "1" and adding gives , exactly the same.

For any T, both right and left hand sides give . - September 14th 2010, 03:07 AMOpalg

The summation is over the integer points in the triangular region in the picture. If you start by fixing y, then you sum over t (along a horizontal line) with t going from y to T (and then you take the results and sum them over y going from 1 to T). But if you start by fixing t, then you sum over y (along a vertical line) with y going from 1 to t (and then you take the results and sum them over t going from 1 to T).

**Edit.**This is just another way of saying what HallsofIvy has explained, but using a graphical picture rather than an algebraic illustration. - September 14th 2010, 08:16 AMleoemil
Brilliant....!