Bascule is reading a book.
What is the probability that the first digit of the page he is on is 1, 2, 3, 4 or 5?
a) obtain an expression
b) approximate a numerical value
(In reply to
re(2): well.. by Cheradenine)
Let's start from first principles. Define a function f(i) as the first digit of i. Define F(i) as 1 if f(i) is in [1, 5] and 0 otherwise. Define g(n) as the sum of all F(i)/n for values of i between 1 and n. If I read you correctly, you're looking for the limit of g(n) as n goes to infinity. If this is not correct, then please let me know where I've gone wrong, and you can disregard the rest of my comment if you so choose.
So what is this limit? If n is of the form 10^x - 1 where x is an arbitrarily large natural number. As x goes to infinity, g(n) = 5/10 + g(n)/10, or g(n) = 5/9. However, consider g(n) where n = 6(10^x) - 1. In this case, g(n) = 5/6 + 1/6 * 5/9 = 45/54 + 5/54 = 25/27.
If the limit of g(n) as n goes to infinity is some value L, some facts must be true. Specifically, for all d > 0, there must be some value N such that |g(n) - L| < d for all n > N. This does not hold true in the problem because for all values of N there is some values 10^x - 1 and 6(10^x) - 1 which are larger than it. These numbers have a large enough interval between them such that there are values for d too small for the limit to work.
What looks like an error to me in your analysis is that you artificially group your sums into octaves demarcated by 10^x. If you try to integrate cos x from zero to infinity, you won't get a defined answer. However, if you try to determine the answer by integrating between zero and multiples of 2pi, you'll always determine that the answer is zero.
re(3): well..
