(Sheet 5) Pudding - Some sequences and series

Here are three unrelated questions, which I hope you will find interesting to ponder.

1. Let \( a_n = (n!)^{\frac{1}{n}} \).  Does the sequence \( (a_n) \) converge as \( n \to \infty \)?
2. Assume that \( a_ n \neq 0 \) for all \( n \) and that \( \frac{1}{a_n} \to 0 \) as \( n \to \infty \).  Does it follow that \( a_ n \to \infty \)?
3. For \( n \geq 1 \), let \( a_n = \frac{1}{\sqrt{n}} + \frac{(-1)^{n-1}}{n} \).  Show that \( a_n > 0 \) and that \( a_n \to 0 \) as \( n \to \infty \).  Show that \( \sum (-1)^{n-1}a_n \) diverges. Why is this interesting?