# My shot at Harmonic Series

To prove Beer Theorem 2 (see “Beer theorems and their proofs“) is to show that the Harmonic Series $1 + {1 \over 2} + {1 \over 3} + ...$ 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 $f$ be a real function such that ${d^2 f} \over {dx^2}$ exists at $x = 0$. Then $\sum\limits_{n=1}^{\infty} f({1 \over n})$ converges absolutely if and only if $f(0) = f'(0)=0$.

Let $f(x) = x$, we have

$\sum\limits_{n=1}^{\infty}f({1 \over n}) = \sum\limits_{n=1}^{\infty}{1 \over n}$,

the Harmonic Series. And,

$f'(x) = {d \over dx} x = 1 \implies f'(0) \neq 0$.

Therefore, by Abu-Mostafa’s theorem, the Harmonic Series diverges.