Question

Find the generating function for the sequence 1,1,1,2,3,4,5,6,....

Answer 1

Answer:

$P(x)=\dfrac{1}{1-x}+\dfrac{x^3}{(1-x)^2} \quad \text{for} \mid x \mid < 1$[/tex]

Step-by-step explanation:

The generating function of a sequence is the power series whose coefficients are the elements of the sequence. For the sequence

$1,1,1,2,3,4,5,6,...$

the generating function would be

$P(x)=1+x+x^2+2x^3+3x^4+4x^5+5x^6+...$

we can multiply P(x) by x to get

$xP(x)=x+x^2+x^3+2x^4+3x^5+4x^6+...$

Note that

$P(x)-xP(x)=1+(2x^3-x^3)+(3x^4-2x^4)+(4x^5-3x^5)+(5x^6-4x^6)+...\\ \\=1+x^3+x^4+x^5+x^6+...=1+x^3(1+x+x^2+x^3+x^4+...)$

which for $\mid x \mid < 1$ can be rewritten as

$(1-x)P(x)=1+\dfrac{x^3}{(1-x)} \quad \Rightarrow \\\\P(x)=\dfrac{1}{(1-x)}+\dfrac{x^3}{(1-x)^2}$