Counting the Digits in a Sequential Integer Series

Last year, I developed a way to count the number of digits in a sequential integer series without having to count the digits in each number. I am still not entirely sure why I pursued this, I think I was working on some programming project that got me thinking about how long a file would be if I wrote sequential numbers to it. Anyway, it was an interesting project; I learned a lot about the summation operation, among other things. Less wonderful was letting this project lay fallow: writing how this equation works has been challenging, to say the least. But, you are probably not really interested in the how, but more of the what I am talking about.