Here's a challenge problem I just made up:

Let $\displaystyle \{a_n\}$ be a sequence of positive real numbers such that $\displaystyle \sum a_n = \infty$. Show that

$\displaystyle \frac{a_1}{a_0}+\frac{a_2}{a_0+a_1}+\frac{a_3}{a_0 +a_1+a_2}+ \dots = \infty$.