Why is the following identity true?
Solution via 'Differences Series' (Or LaTex practice )
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:
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