find constant in upper bound of exponential squence

Question

For some constant $\alpha \in (0,1)$ i would like to find a constant $c_\alpha>0$ such that for all $t\in\mathbb{N}_+$,

$$ \alpha^t\leq c_\alpha t^{-1}. $$

I have no clue here how to start and would be verry glad for any suggestions.

Answer 1

Let $f(t) = t\alpha^t$. You have $$f'(t) = \alpha^t + t\log (\alpha)\alpha^t = 0 \quad \implies \quad t = -\frac1{\log \alpha}.$$ Since $$f''(t) = 2\log(\alpha)\alpha^t + t\log(\alpha)^2\alpha^t = \left(2\log(\alpha)+ t\log(\alpha)^2\right)\alpha^t$$

this proves that $t=-\frac{1}{\log \alpha}$ is a maximu of $f$ on $(0,\infty)$.

So $$f(t) \le f\left(-\frac1{\log\alpha}\right) = -\frac 1{\log \alpha} \alpha^{-\frac{1}{\log\alpha}} = -\frac1{e\log \alpha} = c_\alpha$$