Problem 1108 (difficulty: 9/10)

For all \(\displaystyle k \in \N\) let \(\displaystyle \sum\limits_{n=1}^\infty a^{(k)}_n\) be a divergent series of positive terms. Prove that there is a sequence \(\displaystyle (c_n)\) of positive real numbers such that the series \(\displaystyle \sum\limits_{n=1}^\infty (c_n\cdot a^{(k)}_n)\) are all divergent.

Give me another random problem!