11
$\begingroup$

This is a nice problem on series convergence that I recently stumbled upon. Given a non-negative sequence of real numbers $(a_n)$ such that $$\sum_{n=1}^\infty a_n < \infty,$$ show that there exists a non-decreasing sequence of non-negative numbers $b_n$ such that $$b_n \to \infty \quad\text{ and } \quad \sum_{n=1}^\infty a_n b_n < \infty.$$ In other words, for every convergent series with non-negative terms, there is another convergent series with "substantially larger" terms.

I have a solution (see below), but maybe someone else has a different, simpler, and/or more elegant solution.

  • 0
    I see a series with terms $a_n$, and a summation over $k$. Shouldn't the terms be $a_k$ to match the summation? Or should the summation be $\displaystyle\sum_{n=1}^\infty$? I wonder what I'm missing here...2012-10-19
  • 0
    You're welcome. This is quite an interesting find you have!2012-10-20

2 Answers 2