Consider an integer with $N$ digits (in decimal notation, without leading zeroes) $D_1, D_2, D_3, \ldots , D_N$. Here, $D_1$ is the most significant digit and $D_N$ the least significant. The weight of this integer is defined as
$$\sum_{i=2}^N (D_i - D_{i-1})$$
Could you help me convert the summation to integral? I need to find the weight of the number here. Thanks Edit N tends to infinity