Hi guys,

I was hoping someone would be able to tell me the name of this technique (described below) I used so I can go away and do some reading about it and hopefully learn it in a more structured way. Also if there are any similar techniques for deducing polynomials from series (like expanding the series) could you tell me the names of them so I can do some reading about them too. Or any other techniques for finding the sum of squares. Thanks.

The technique:

So I was trying to deduce the sum of squares:

$\displaystyle \sum_{x=1}^{n}x^2=x^3/3+x^2/2+x/6$

and I used the technique where you go down the levels of the difference. i.e

the first few terms of this series is:

1,5,14,30,55,91,140

difference of these terms is:

4,9,16,25,36,49

and the diff. of the diff.:

5,7,9,11,13

and the level 3 diff.:

2,2,2,2

so now we can see that since it's level 3 .

$\displaystyle \sum_{x=1}^{n}x^2=O(x^3)$

comparing this to $\displaystyle x^3$ series:

series:

1,8,27,64,125,216,343

level 1:

7,19,37,61,91,127

level 2:

12,18,24,30,36

level 3:

6,6,6,6

comparing level 3's of $\displaystyle x^3$ and $\displaystyle \sum^n_{x=1}x^3 $ we get:

$\displaystyle \sum_{x=1}^{n}x^2=x^3/3+O(x^2)$

Now the level 2 difference of $\displaystyle \sum_{x=1}^{n}x^2-x^3/3$ is:

1,1,1,1,1

and level 2 difference for $\displaystyle x^2$ is:

2,2,2,2,2

so we now have:

$\displaystyle \sum_{x=1}^{n}x^2=x^3/3+x^2/2+O(x)$

Finally the level 1 difference of $\displaystyle \sum_{x=1}^{n}x^2-x^3/3-x^2/2$ is:

1/6,1/6,1/6,1/6,1/6,1/6

and for $\displaystyle x$ is:

1,1,1,1,1,1

$\displaystyle \sum_{x=1}^{n}x^2=x^3/3+x^2/2+x/6$

and I got the right result. Now what's this technique called?