# Number Of Digits In Powers Of Two

Calculus Level 3

Let $$a_k$$ denote the number of digits in $$2^k$$. For instance, since $$2^4 = 16$$, we have $$a_4 = 2$$.

Compute the limit: $\lim_{n\to\infty} \frac{1}{n^2} \sum_{k=1}^{n} a_k$

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