Inequality

Algebra Level 4

If aia_i (i=1,2,,n)(i=1, 2, \ldots, n) is a sequence of real numbers satisfying i=1nai9i5 \displaystyle \sum_{i=1}^n a_i 9^i \leq 5 , what is the smallest possible constant kk such that limni=1nai2ik?\lim_{n\to \infty} \displaystyle \sum_{i=1}^n |a_i| 2^i \leq k?

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