Generating function - $1,1,1,1,1,1$

Question

What is the generating function for $1,1,1,1,1,1$?

I know this to be $1 + x +x^2+ x^3+x^4+x^5$

But then I saw this:

$$\frac{x^6-1}{x-1} = 1 + x +x^2+ x^3+x^4+x^5$$

How was this equality obtained?

Was it just a random (manual)? or is there any method involved to obtain that fractional part?

Are you asking why $\frac{x^6-1}{x-1}=x^5+x^4+x^3+x^2+x+1$? Multiply both sides by $x-1$, expand the RHS, and things should cancel out to $x^6-1$. In general, $x^a-1=(x-a)(x^{a-1}+x^{a-2}+\dots+x^2+x+1)$ — user574848, Dec 25 '18 at 10:59
thanks user574848, but why there was a need to write that equation ? — swapnil, Dec 25 '18 at 11:07
okay got it, that was a silly doubt didn't notice that there is geometric progression involved. I though it must be related with theorem related to the generating function. — swapnil, Dec 25 '18 at 11:21

score 0 · Answer 1 · answered Dec 27 '18 at 05:55

0

As noted in the comments to your question, the equation

$$\frac{x^6 - 1}{x-1} = 1 + x + x^2 + x^3 + x^4 + x^5$$

comes about as the sum of a finite geometric series. Suppose we have a finite geometric series of ratio $x$. Then it can be shown

$$1 + x + x^2 + ... + x^n = \frac{x^{n+1} - 1}{x-1}$$

Take $n=5$ and the equality results.

answered Dec 27 '18 at 05:55

PrincessEev

It should be stated when using the sum of the geometric power that it doesn't hold when $x = 1$. – Jan 08 '19 at 05:36

1 Answers1