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Let $\alpha = n^{c_1}$ and $\beta = n^{c_2}$. Is there a good upper bound (an asymptotic bound would be good as well) for $f(n) = (1 - \frac{1}{\alpha})^\beta$. I am particularly interested in a bound for the cases when $c_2 < c_1$, if at all that matters.

Any ideas and/or references would be very much appreciated.

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Certainly $f(n)\le1$. If $c_2\lt c_1$, then $\lim_{n\to\infty}f(n)=1$, so $1$ is looking like a pretty good upper bound.

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