Step by step solution to "Determine the convergence or divergence of the following..."

melodykap

Answered question

2021-08-19

Determine the convergence or divergence of the following sequences. If it converges give it its limit

a_{n} = \frac{\ln (n^{3})}{2 n}

Answer & Explanation

unett

Skilled2021-08-20Added 119 answers

Given the $n^{t h}$ term of a sequence $a_{n} = \frac{\log (n^{3})}{n} = \frac{3 \log (n)}{n}$
To check the convergence of the sequence $lim_{n \to \infty} a_{n}$ should be finitely defined
Now find $lim_{n \to \infty} a_{n}$ for the given sequence.
$lim_{n \to \infty} a_{n} = lim_{n \to \infty} \frac{3 \log n}{n} = \frac{\infty}{\infty}$ (Indeterminate form)
Using L'Hopital rule to solve the limit.
$lim_{n \to \infty} a_{n} = lim_{n \to \infty} \frac{3 \frac{d (\log n)}{d n}}{\frac{d (n)}{d n}}$
$lim_{n \to \infty} a_{n} = lim_{n \to \infty} \frac{\frac{3}{n}}{1} = \frac{3}{\infty}$
$lim_{n \to \infty} a_{n} = 0$
As the value of limit is zero. Which is finitely defined. Hence the given sequence is convergent with $lim_{n \to \infty} a_{n} = 0$

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