Is this function bounded? $g(n)=t>1\text{ s.t. } \int_1^n (1-i^{-p})^t i^{-p} \, di =\varepsilon$ Let $p>1,t>1,\varepsilon>0$. Assuming the function below exists, when is it bounded?
$$g(n)=t>1\text{ s.t. } \int_1^n (1-i^{-p} )^t i^{-p} \,di =\varepsilon$$
From simulations I suspect it's bounded for all $p>1$ and $\varepsilon>0$, can this be shown analytically?
For instance, for $p=2$ and $\varepsilon=0.01$ the graph obtained numerically


It seems that you're just asking if $$\lim_{C \to +\infty} \int_1^\infty (1-x^{-p})^Cx^{-p}dx = 0$$ for any fixed $p > 1$. And this is indeed the case, by dominated convergence theorem.
