<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] One way is to look at 
$$
1+2x^2+3x^4+4x^6+5x^8+\dots
$$
as
$$
1+2t+3t^2+4t^3+5t^4+\dots
$$
where $t=x^2$. The last series is the derivative of
$$
1+t+t^2+t^3+t^4+t^5+\dots=\frac1{1-t}
$$
Therefore,
$$
1+2t+3t^2+4t^3+5t^4+\dots=\frac1{(1-t)^2}
$$
and
$$
1+2x^2+3x^4+4x^6+5x^8+\dots=\frac1{(1-x^2)^2}
$$
Now, just multiply by $1+x$:
$$
\begin{align}
1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8+5x^9+\dots
&=\frac{1+x}{(1-x^2)^2}\\
&=\frac1{(1-x)(1-x^2)}
\end{align}
$$
</s>