Question

Determine if the sequence converges: [ln(n)]^2 /n

Answer 1

We will use l´Hopital´s rule for calculating limits involving indeterminate form (in this case: ∞ / ∞ ) using the derivative of the numerator and denominator:$\lim_{n \to \infty} \frac{ln^{2}n }{n} = \lim_{n \to \infty} \frac{2ln(n)* \frac{1}{n} }{1} = \lim_{n \to \infty} \frac{2ln(n)}{n}$ This is still form ∞/∞ and we will use the derivative again:$\lim_{n \to \infty} \frac{2ln(n)}{n} = \lim_{n \to \infty} \frac{ \frac{1}{n} }{1} = \lim_{n \to \infty} \frac{1}{n}$=1/∞ = 0
The sequence converges.