Why is the following identity true?

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- Jul 17th 2011, 02:04 PMzhangvictSummation identity
Why is the following identity true?

- Jul 17th 2011, 02:07 PMSironRe: Summation identity
There's nothing to see? ...

- Jul 17th 2011, 02:10 PMzhangvictRe: Summation identity
sorry, had problems posting the equation. It is there now

- Jul 17th 2011, 02:26 PMpickslidesRe: Summation identity
The best way to answer this 'why?' is to prove it yourself.

Do you know of mathematical induction? - Jul 17th 2011, 02:41 PMAlso sprach ZarathustraRe: Summation identity

Solution via 'Differences Series' (Or LaTex practice (Hi))

Step no. 0:

Step no. 1:

Step no. 2:

Step no. 3:

Step no. 4:

Now we find the generating function from 4 and up...

The generating function of #4 is

The generating function of #3 is

The generating function of #2 is

The generating function of #1 is

The generating function of #0 is

We get that generating function of .

So, is the coefficient of :

Hence the coefficient of is:

The conclusion is:

- Jul 17th 2011, 03:11 PMArchie MeadeRe: Summation identity
There are a number of ways to show why the formula holds.

One easy method is to use the fact that there is a relationship

between the sum of positive natural numbers

and the sum of their squares....

as it is a very basic arithmetic series.

Hence, induction can prove without doubt that

- Jul 17th 2011, 09:14 PMProve ItRe: Summation identity
http://i22.photobucket.com/albums/b3...mofsquares.jpg

You will also need this...

http://i22.photobucket.com/albums/b3...ofintegers.jpg