arrow_back
We can think of the set of all `i,j` between `0` and `n`
as the shaded region on the grid:

Then the set of pairs that we sum over to get the sum of the `c _{k}`'s is:

The remaining terms are the summation over the region:

We are trying to show that as `n` grows, the summation
over this region goes to zero. The trick is to notice that although the
region grows in size as `n` grows, the terms that belong to the
region lie further and further out. We shall show that the upper
triangular region lies
inside the larger "L-shaped" region shown below, and make an estimate of the summation
over that region.

(Notice for the estimate below that this last shape is obtained
by taking a square of side `n` and deleting a square of side
`n/2` from the corner.)