<s>[INST] How can I write this power series as a power series representation? How can I write this power series ($1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8....$) as a power series representation (like $\dfrac{1}{1-x}$ or something neat like that)? 
 [/INST] Hint: using $y=x^2$ and derivative in $y$: $$(1+x)(1+2x^2+3x^4+\ldots) $$
$$ =(1+x)(1+2y+3y^2+4y^3 +\ldots)$$
$$= (1+x)(y+y^2+y^3+y^4+\ldots)'$$
$$ = (1+x)\left( \frac{y}{1-y}\right)'$$
Edit:
$$ = (1+x) \frac{1}{(1-y)^2} $$
$$ = \frac{1+x}{(1-x^2)^2}  $$
$$ = \frac{1}{(1-x)(1+x^2)}.$$
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