<s>[INST] What is the limit of the serie $\sum_{k=0}^{\infty}\frac{x^4}{(1+x^2)^k}$ Find the limit of the series :

$$\sum_{k=0}^{\infty}\frac{x^4}{(1+x^2)^k}$$ 

So, I applied the ratio test and did $\frac{x^4}{(1+x^2)^{n+1}}$*$\frac{(1+x^2)^n}{x^4}$.
Then, I get $1 + \frac{1}{x^2}$. I then multiply $1 + \frac{1}{x^2}$ by $x^4$, to which I get an answer choice of (E) $x^4 + x^2$
I kind of want to know if I'm actually doing the process correctly, and that $x^4 + x^2$ is the limit of the aforementioned series.
These are the following answer choices:
(A) $x^6 + x^4$
(B) $\frac{x^6}{1+x^2}$
(C) $x^6$
(D) $x^4$ + $\frac{x^6}{1+x^2}$
(E) $x^4$ + $x^2$
 [/INST] hint: You basically have a geometric series with the first term $a_1 = x^4, r = \dfrac{1}{1+x^2}$. The sum is $\dfrac{a_1}{1-r}=...$
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