Given $|a-b|<\epsilon$ and $|a|\le K$; $|b|\le K$ give an estimate for the approximation $a^n \approx b^n$, where $n$ is a positive integer.
We have that $|a^n-b^n|<\epsilon_0$; $|a|^n\le K^n$; $|b|^n\le K^n$
$|a^n-b^n| = |a^n(1-\frac{b^n}{a^n})|=|a^n||1-(\frac{b^n}{a^n})|\le K^n|1-(\frac{b}{a})^n|$
$|a^{n}-b^{n}|=|(a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})| \leq \epsilon nK^{n-1}$