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all, suppose that $$ \lim_{t\rightarrow \infty} r(t)=r_0>0, $$ I am wondering whether there exists $C>0$ and $\alpha>0$ such that $$ |r(t)-r_0|\leq C(t+1)^{-\alpha}, \quad \forall t>0. $$

If it were the case, then algebraic convergence is the slowest convergence。

I thought the above claim was not right, but I did not find such a supporting example. Any help will be greatly appreciated.

Teh
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  • Dear Guys, I found an example “algebraic-algebraic”:$r(t)=(1+\frac{1}{t})^{\frac{1}{t}}$ – Teh Dec 02 '16 at 12:40

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We may consider $\lim r(t) =0$, because then $\lim (r(t)+r_0)$ will tend to $r_0$.

If we do not believe the statement, we should try to find a counter example, that is we need to find a function $r(t)$ such that $$\lim_{t\to\infty}r(t)=0$$ while $$t^\alpha r(t)$$ is unbounded for all $\alpha$.

Does this help?

AD - Stop Putin -
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