1. ## Compare A ,B

Hello :

Compare A ,B:

2. To potential helpers : please read this post : "http://www.mathhelpforum.com/math-help/number-theory/93501-facto-sum.html "

3. Hello, dhiab!

Compare $\displaystyle A\text{ and }B\!:\quad\begin{array}{ccc} A &=& \dfrac{1.0000004}{(1.0000006)^2} \\ \\[-3mm] B &=& \dfrac{(0.9999995)^2}{0.9999998} \end{array}$

$\displaystyle \text{Let }d \:=\:\frac{a}{10^7},\:\text{where }a\text{ is a digit } \leq 6$

Then: .$\displaystyle \begin{array}{ccc}A &\approx & \dfrac{1+d}{(1+d)^2} \\ \\ [-3mm]B & \approx& \dfrac{(1-d)^2}{1-d} \end{array}$

We have: . $\displaystyle \overbrace{\frac{1+d}{(1+d)^2}}^{A} \quad^{>}_{<}\quad \overbrace{\frac{(1-d)^2}{1-d}}^{B}$

Then: . $\displaystyle (1+d)(1-d)\quad ^{>}_{<}\quad (1-d)^2(1+d)^2$

. . . . . . . . . . $\displaystyle 1 - d^2 \quad ^{>}_{<}\quad (1-d^2)^2$

. . . . . . . . . . $\displaystyle 1 - d^2 \quad ^{>}_{<}\quad 1 - 2d^2 + d^4$

. . . . . . . . . . . . $\displaystyle d^2\quad ^{>}_{<}\quad d^4$

Divide by positive $\displaystyle d^2\!:\quad 1 \quad ^{>}_{<}\quad d^2$ .[1]

Since $\displaystyle d < 1$, then $\displaystyle d^2 < 1$

Hence, [1] becomes: .$\displaystyle 1 \;\;{\color{red}>}\;\;d^2$

Therefore: .$\displaystyle A \;{\color{red}>}\;B$

4. Originally Posted by Bruno J.