Part 3/6:
\lim_{n \to \infty} \frac{\log n}{n} \rightarrow \lim_{n \to \infty} \frac{\frac{d}{dn}(\log n)}{\frac{d}{dn}(n)} = \lim_{n \to \infty} \frac{\frac{1}{n}}{1}
]
This leads to:
[
\lim_{n \to \infty} \frac{1}{n} = 0
]
Thus, it's confirmed that ( \lim_{n \to \infty} \frac{\log n}{n} = 0 ).
Convergence of Alternating Sequences
Consider the sequence defined by ( a_n = (-1)^n ). As ( n ) alternates between even and odd, the sequence oscillates between 1 and -1. Graphically, this can be represented as points alternating above and below the x-axis, clearly indicating no approach towards a unique limit. Therefore, we conclude:
[
\lim_{n \to \infty} a_n \text{ does not exist.}
]