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- TL;DR Summary
- The limit of the sequence ##a^n/n## for ##a>1## is ##+\infty## but how to prove it without reverting to limit of functions and using L'Hopital rule.
We have the limit of the sequence ##\frac{a^n}{n}## where ##a>1##. I know it is ##+\infty## and i can prove it by switching to the function ##\frac{a^x}{x}## and using L'Hopital.
But how do i prove it using more basic calculus, without the knowledge of functions and derivatives and L'Hopital.
I can prove that the sequence is increasing (after a certain ##n_0##) but have trouble proving that the sequence is not bounded. More specifically
$$\frac{a^n}{n}>M\Rightarrow n\ln a-\ln n>\ln M$$ and where do i go from here?
But how do i prove it using more basic calculus, without the knowledge of functions and derivatives and L'Hopital.
I can prove that the sequence is increasing (after a certain ##n_0##) but have trouble proving that the sequence is not bounded. More specifically
$$\frac{a^n}{n}>M\Rightarrow n\ln a-\ln n>\ln M$$ and where do i go from here?
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