My shot at Harmonic Series

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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.

1 thought on “My shot at Harmonic Series

  1. Pingback: Beer theorems and their proofs | Vroom

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