<s>[INST] Compare $A=\frac{1.0\,000\,004}{(1.0\,000\,006)^2}$ and $ B=\frac{(0.9\,999\,995)^2}{0.9\,999\,998}$ My work:


*

*$1.0\,000\,004 = 1+\frac{4}{10^7}=1+\frac{1}{125\cdot 10^6}$

*$ (1.0\,000\,006)^2=(1+\frac{6}{10^7})^2=(1+\frac{3}{5\cdot 10^6})^2$

*$ (0.9\,999\,995)^2=(\frac{9\,999\,995}{10^7})^2=(\frac{1\,999\,999}{2\cdot10^6})^2$

*$ 0.9\,999\,998=\frac{9.999\,998}{10^7}=\frac{4\,999\,999}{5\cdot10^6}$
The two fractions become:
$A=\dfrac{1+\frac{1}{125\cdot 10^6}}{(1+\frac{3}{5\cdot 10^6})^2}$ and $B=\dfrac{(\frac{1\,999\,999}{2\cdot10^6})^2}{\frac{4\,999\,999}{5\cdot10^6}}$
At this stage, I don't know how to continue.
Any help is appreciated. Thank you.
 [/INST] $$A=\frac {10^7\times (10^7+4)}{(10^7+6)^2}$$
$$B=\frac {(10^7-5)^2}{10^7\times (10^7-2)}$$
$$\frac AB=\frac {10^{14}(10^7+4)(10^7-2)}{(10^7+6)^2(10^7-5)^2}$$
The numerator is $10^{28}+2\cdot10^{21}-8\cdot10^{14}$
The denominator is $10^{28}+2\cdot 10^{21}-59\cdot10^{14}-60\cdot10^7+900$
It is easy to see from this which is larger.
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