Answer :
Answer:
[tex]P(x)=\dfrac{1}{1-x}+\dfrac{x^3}{(1-x)^2} \quad \text{for} \mid x \mid < 1[/tex][/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
[tex]1,1,1,2,3,4,5,6,...[/tex]
the generating function would be
[tex]P(x)=1+x+x^2+2x^3+3x^4+4x^5+5x^6+...\\[/tex]
we can multiply P(x) by x to get
[tex]xP(x)=x+x^2+x^3+2x^4+3x^5+4x^6+...[/tex]
Note that
[tex]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+...)[/tex]
which for [tex]\mid x \mid < 1[/tex] can be rewritten as
[tex](1-x)P(x)=1+\dfrac{x^3}{(1-x)} \quad \Rightarrow \\\\P(x)=\dfrac{1}{(1-x)}+\dfrac{x^3}{(1-x)^2}[/tex]