How to prove this approximation?

[mfb]

@stevendaryl: The formula is a good approximation even in a range where the approximations you used are no longer valid. Proving that doesn't work with your approach. Which is correct, but not general enough.

 

[stevendaryl]

@stevendaryl: The formula is a good approximation even in a range where the approximations you used are no longer valid. Proving that doesn't work with your approach. Which is correct, but not general enough.

His formula is the same as mine, after substituting $x_2 \Rightarrow log(x_2)$, $x_1 \Rightarrow log(x_1)$, $n \Rightarrow n+1$. So if my formula is good as long as $\frac{x_2}{x_1} < K$, then the replacement would extend that range to $\frac{x_2}{x_1} < (x_1)^K$.

 

[mfb]

That is still not sufficient to cover the range where the original formula is a good approximation.