Number Of Digits In Powers Of Two

Calculus Level 2

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

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