If $a_i$$(i=1, 2, \ldots, n)$ is a sequence of real numbers satisfying $\displaystyle \sum_{i=1}^n a_i 9^i \leq 5$, what is the smallest possible constant $k$ such that $\lim_{n\to \infty} \displaystyle \sum_{i=1}^n |a_i| 2^i \leq k?$

Your answer seems reasonable.
Find out if you're right!