Number Of Digits In Powers Of Two

Calculus Level 2

Let aka_k denote the number of digits in 2k2^k. For instance, since 24=162^4 = 16, we have a4=2a_4 = 2.

Compute the limit

limn1n2k=1nak.\lim_{n\to\infty} \frac{1}{n^2} \sum_{k=1}^{n} a_k.

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