To prove Beer Theorem 2 (see “Beer theorems and their proofs“) is to show that the Harmonic Series diverges.
Below is my shot at it.
Yaser S. Abu-Mostafa proved a theorem in an article titled “A differentiation test for absolute convergence” (see Mathematics Magazine 57(4), 228-231)
His theorem states that
Let be a real function such that exists at . Then converges absolutely if and only if .
Let , we have
the Harmonic Series. And,
Therefore, by Abu-Mostafa’s theorem, the Harmonic Series diverges.