I have this function at hand with 3 non-negative integer parameters that I would like to put a bound to:

$$f(n, b, d) = 1 - \left(1 - \left(\frac1n\right)^{b+d-1}\right)^{\left(n^{b-1}\right)}$$

My end goal is to show that for any positive $\varepsilon_0$ there is a $n_0$ and $d^*$ such that for all $n \ge n_0$ and $b > 0$, $f(n, b, d^*) < \varepsilon_0$. In simpler terms, I want to show that $f$ can be made arbitrarily small for any $n$ larger than a constant by affixing $d$ to be another large-enough constant.

To do this, I am looking for a decreasing function $g(n, d)$ which satisfies $f(n, b, d) \le g(n, d)$ for all $b$. Playing around with $f$ in Desmos's graphing calculator, it seems that $g(n, d) = \left(\frac{2}{d}\right)^d$ is such a function and seems very tight. link to graph

I don't need it to be tight. I can make do with any other decreasing function that allows me to reach my goal. (I can even do away with the bounding function altogether so long as I can reach my goal, but I still am curious about how one can bound a function like $f$.)



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