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Determine the convergence or divergence of f(n) = (loga n) / n

by RoRi

Consider the function f(n) defined by

    \[ f(n) = \frac{\log_a n}{n}, \qquad a > 1. \]

Determine whether the sequence \{ f(n) \} converges or diverges, and if it converges find the limit.

First, we recall the definition of \log_a n,

    \[ \log_a n = \frac{\log n}{\log a}. \]

Therefore,

    \begin{align*} f(n) &= \frac{\log_a n}{n} \\[9pt] &= \frac{ \frac{\log n}{\log a}}{n} \\[9pt] &= \left( \frac{\log n}{n} \right) \left( \frac{1}{\log a} \right). \end{align*}

By property (10.11) on page 380 of Apostol we know

    \[ \lim_{n \to \infty} \frac{\log n}{n} = 0. \]

Since \frac{1}{\log a} is just a constant we then have

    \[ \lim_{n \to \infty} f(n) = 0. \]

Hence, \{ f (n) \} converges to the limit 0.

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