melodykap

2021-08-19

Determine the convergence or divergence of the following sequences. If it converges give it its limit
${a}_{n}=\frac{\mathrm{ln}\left({n}^{3}\right)}{2n}$

unett

Given the ${n}^{th}$ term of a sequence ${a}_{n}=\frac{\mathrm{log}\left({n}^{3}\right)}{n}=\frac{3\mathrm{log}\left(n\right)}{n}$
To check the convergence of the sequence $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}$ should be finitely defined
Now find $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}$ for the given sequence.
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{3\mathrm{log}n}{n}=\frac{\mathrm{\infty }}{\mathrm{\infty }}$(Indeterminate form)
Using L'Hopital rule to solve the limit.
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{3\frac{d\left(\mathrm{log}n\right)}{dn}}{\frac{d\left(n\right)}{dn}}$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{3}{n}}{1}=\frac{3}{\mathrm{\infty }}$
$\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=0$
As the value of limit is zero. Which is finitely defined. Hence the given sequence is convergent with $\underset{n\to \mathrm{\infty }}{lim}{a}_{n}=0$

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