<s>[INST] Finding a Function That Approaches Another Function One of my math professors gave me the following challenge. It isn't graded, it's just for fun.
Consider the function:
\begin{equation*}
f_n(x)=x+3^3x^3+5^3x^5+...+(2n-1)^3x^{2n-1},~x \in (0, 1).
\end{equation*}
I want to find which of the following functions $f_n$ is getting close to as $n$ gets larger:
$ \displaystyle a)\frac{x(x+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4}$ $\displaystyle b) \frac{x(x^2+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4}$
$\displaystyle c) \frac{x^2(x+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4}$ $\displaystyle d) \frac{x^2(x^2+1)(x^4+22x^2+1)}{(x-1)^4(x+1)^4}$
Based on some tests i ran in mathematica by giving $n$ and $x$ values, it looks like $b)$ is the answer, but I am not sure. Can anyone confirm or deny this, and show how one might find the right answer, either with pen and paper or by using mathematica or maple or some other software?
 [/INST] Yes looks like it must be b)
Consider the following:
There is only one $x$ factor, so this eliminates choices c) and d).
The function is odd, i.e $f(-x) = -f(-x)$, so this eliminates a).
Thus b) must be the answer.
To find the answer by hand you can do the following:
Start with 
$$f(x) = x + x^3 + x^5 + \cdots = \frac{x}{1-x^2} $$
Now differentiate 
$$f'(x)  = 1 + 3x^2 + 5x^4 + \cdots $$
Multiply by $x$
$$f'(x)x = x + 3x^3 + 5x^5 + \cdots $$
Differentiate again
$$(f'(x)x)' = 1 + 3^2 x^2 + 5^2 x^4 + \cdots $$
Multiply by $x$, differentiate again and then multiply by $x$ to give the answer.
</s>