Essentially the title is the full question. In order to do some other questions I am trying to learn how to naturally figure out series representations of functions. 

Computing the Taylor series of this function
$ \frac{1}{\sqrt{1+t^2}} = 1- \frac{x^2}{2}+\frac{3x^4}{8}-\frac{5x^6}{16}+\frac{35x^8}{128}+... $ 

However when trying to see this in series representation I struggle.. I can see that there is a $\sum^{\infty}\_{n=0} \frac{(-1)^nx^{2n}}{(2n)!}$ factor.

In fact taking a look now the $(2n)!$ can't be there as it doesnt work with my series... and also how could I include factors that create the numerators? 

Perhaps there is a pattern I should be noticing here but I am struggling.. 

Any help is appreciated.