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

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