We have all seen this easy exercise: show that where is a real constant, i.e. is independent of The proof is quite simple: we have
But things get interesting when depends on
For example, what is Well, the answer is and that’s a trivial consequence of a much nicer result that we are going to prove in Problem 2. But first, we need to prove something, which is nice by itself.
Problem 1. If is a real constant, show that
Solution. Choose an integer such that Then
Now the result follows from the facts that
where is the Euler’s constant, and
Problem 2. Show that
Solution. Let By the Example in this post, we have
On the other hand, again by we have
So we have proved that
which completes the solution because and, by Problem 1, and so, by the squeeze theorem,
Remark. There is another proof of Problem 2 that uses Tannery’s theorem.
Example. Show that where are Bernoulli numbers.
Solution. In the problem in this post, put to get
Thus, since and for we have
and the result follows because, by Problem 2,
Exercise 1. If is a real constant, what is
Exercise 2. Show that without using the result given in Problem 2.
Hint. You’ll only need the easy half of the solution of Problem 2.