Is there any generic lower bound of $(1-x)^n$ and upper bound of $(1+x)^n$, where $0<x<1$ and $n\in\mathbb{N}$?
2026-04-07 05:06:26.1775538386
Upper bound of $(1+x)^n$ and lower bound of $(1-x)^n$
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$(1-x)^n> 0^n=0$ for $0<x<1$. Moreover, for any $M>0$, \begin{align} (1+x)^n>M \iff\\ n> \frac{\ln M}{\ln(1+x)}. \end{align} Of course, such an $n$ can always be chosen. It should be clear that $(1+x)^n$ is unbounded from above.