The sequence \(\left<log\dfrac{1}{n}\right>\) is

Concept:

The N^th term test

If \(\lim_{n\rightarrow ∞ }\left ( \sum_{n=0}^{∞ }a_{n} \right )=L\), where L is any tangible number other than zero. Then, \(\left ( \sum_{n=0}^{∞ }a_{n} \right )\) diverges.

This is also called the Divergence test.

Calculation:

We have, \(\left<log\dfrac{1}{n}\right>\)

\(\Rightarrow \sum_{n=1}^{∞ } log\left ( \frac{1}{n} \right )\)

\(\Rightarrow \lim_{n \to ∞ }\left [\sum_{n=1}^{∞ } log\left ( \frac{1}{n} \right ) \right ]\)

\(\Rightarrow \lim_{n \to ∞ }log\left ( \frac{1}{n} \right )\)

So, as n → ∞, \(\frac{1}{n}\) → 0

\(\Rightarrow \lim_{n \to ∞ }log\left ( \frac{1}{n} \right ) = -∞ \neq 0\)

Thus, our series diverges to -∞ by the n^th term test.

Hence, The sequence \(\left<log\dfrac{1}{n}\right>\) is divergent to -∞.