How to write the Taylor polynomial(not series) of the function $\frac{1}{1+(x)^2} $

Question

I tried using the expansion of $\frac{1}{1-x} $ and then substitute $-x^2$ for every $x$. But then I realised that I have to find the partial sum for some ($n$) not the infinite.I thought to first find the infinite sum and then make it into a partial by simply "changing" the infinity with $(n)$.Is that correct? and if not how can I find it?

Answer 1

Let use the binomial binomial series expansion with $n=-1$ and $y=x^2$

$$(1+y)^n=1+ny++\frac{n(n-1)}{2}y^2+...+\binom{n}{k}y^k+o(y^k) \quad y\to0$$

that is

$$(1+x^2)^{-1}=1-x^2+x^4-x^6+x^8...+(-1)^{k}x^{2k}+o(x^{2k})$$