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) (ln^(2)n )/(n) = \lim_(n \to \infty) (2ln(n)* (1)/(n) )/(1) = \lim_(n \to \infty) (2ln(n))/(n)` This is still form ∞/∞ and we will use the derivative again:
`\lim_(n \to \infty) (2ln(n))/(n) = \lim_(n \to \infty) ( (1)/(n) )/(1) = \lim_(n \to \infty) (1)/(n)`=1/∞ = 0
The sequence converges.