Landau Notation Problem

I have this function$$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$$$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \frac{1}{(1+t^2)^{n}}\leq\frac{1}{(1+t)^{n}} \\ \text{ We got } K_n=\int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt\leq\frac{1}{n-1}\times\frac{1}{2^{n-1}}$$I need to prove that $$K_n=O(\frac{1}{n2^{n}})$$ I would like to have a detailed justification please or hint on how I can find the answer. My definition of O notation is that let's assume that $a_n, b_n, m_n$ positives sequences such that $m_n$ is bounded. We says that $a_n=O(b_n) \Leftrightarrow \exists N, \forall n\geq N, a_n\leq b_nM_n $.