Problem 370 (difficulty: 5/10)

(a) Prove that if \(\displaystyle \liminf{\log {1\over a_k} \over \log k}>1\), then \(\displaystyle \sum a_k\) is convergent.

(b) Prove that if \(\displaystyle \limsup{\log {1\over a_k} \over \log k}<1\), then \(\displaystyle \sum a_k\) is divergent.

(c) Construct a sequence \(\displaystyle a_n\) such that \(\displaystyle {\log {1\over a_k} \over \log k}\to1\), and \(\displaystyle \sum a_k\) convergent.

(d) Construct a sequence \(\displaystyle a_n\) such that \(\displaystyle {\log {1\over a_k} \over \log k}\to1\), and \(\displaystyle \sum a_k\) divergent.

Give me another random problem!