Suppose $a>0$, define sequence $a_n$ by $$a_n=\sqrt{n}\left(\sqrt[n]{ea}-\sqrt[n]{a}\right).$$ Find $\lim_{n\to \infty} a_n$ if it exists.
By observing the speed of $\left(\sqrt[n]{ea}-\sqrt[n]{a}\right)$ I thought that limit is $0$.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
$a_n=a^{1/n} \sqrt n (e^{1/n}-1)$. Now $a^{1/n} \to 1$ and $e^{1/n}-1 \sim \frac 1 n$ as $n \to \infty$. Hence the limit is $0$.
answered Jun 18, 2020 at 12:26