How is $x^2+1=(1/{x^2})[1-{1}/{x^2}+{1}/{x^4}-{1}/{x^6}+\cdots]$? The author of my book writes:
$$x^2+1=x^2\left(1+\frac{1}{x^2}\right)$$
$$=\frac{1}{x^2}\left[1-\frac{1}{x^2}+\frac{1}{x^4}-\frac{1}{x^6}+\cdots\right]$$
I do not understand the last step. How did the author write the last step. Please help.

You may write $$x^2 + 1 = x^2(1 + \frac{1}{x^2})$$
as $$\frac{1}{(1 + \frac{1}{x^2})} = \frac{x^2}{1 + x^2}$$
and 
$$\frac{1}{1 + \frac{1}{x^2}}  = \sum_{n=0}^{\infty} \frac{(-1)^n}{x^{2n}}$$

Edit: To avoid any more downvotes, the author has made a mistake, clearly the equation does not hold, this was just an attemptive to reinterpret whatever the author's intention was.  

