Problem 1116 (difficulty: 5/10)

(a) Show that if \(\displaystyle \displaystyle \Olim \left( \big|a_n\big|^{\frac1{\log n}} \right) < \frac1e\), then \(\displaystyle \displaystyle\sum_{n=1}^\infty a_n\) is absolutely convergent.

(b) Show that if \(\displaystyle a_n\ge0\) and \(\displaystyle \displaystyle \Ulim \left( \big|a_n\big|^{\frac1{\log n}} \right) > \frac1e \), then \(\displaystyle \displaystyle\sum_{n=1}^\infty a_n\) is divergent.

(c) Can any conclusions be made about the convergence of \(\displaystyle \displaystyle\sum_{n=1}^\infty a_n\) if \(\displaystyle a_n>0\) and \(\displaystyle \displaystyle \lim \left( \big|a_n\big|^{\frac1{\log n}} \right) =\frac1e \)?

Give me another random problem!