Inequality

Algebra Level 4

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 $\lim_{n\to \infty} \displaystyle \sum_{i=1}^n |a_i| 2^i \leq k?$

×