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