Is there any way to re-write the following
$$p_{\hat y}
\simeq\frac{1}{N-\frac{1}{T}U_1(x)+\frac{1}{2T^2}U_2(x)}$$
such that
$$p_{\hat y}\propto \left(U_1-\frac{U_2}{2T}\right)/T $$
$N$ is a positive integer number, $U_1$ and $U_2$ are functions of $x$ but they are always positive, $T$ is a parameter ranging in $[0,+\infty]$.
I found this on a paper, but I cannot understand why the signs of $U_1$ and $U_2$ change from minus to plus and vice-versa. Maybe there is some fractional identity that I forgot about. Thanks.
Best Answer
What it looks to me is that this is the result of a Taylor expansion for large values on $N$. $$\frac{1}{N-\frac{1}{T}U_1+\frac{1}{2T^2}U_2}=\frac{1}{N}+\frac{\frac{U_1}{T}-\frac{U_2}{2 T^2}}{N^2}+O\left(\frac{1}{N^3}\right)$$ that is to say $$\frac{1}{N}+\frac{1}{N^2}\left(U_1-\frac{U_2}{2T}\right)\frac 1T+\cdots$$