Question-216722

Question Number 216722 by issac last updated on 17/Feb/25

Answered by MrGaster last updated on 17/Feb/25

\int_{0}^{\infty} (\frac{1}{ℓ} - \frac{1}{p}) d ℓ = \int_{0}^{\infty} (\frac{p - ℓ}{ℓ p}) d ℓ

{\int_{0}^{\infty} (\frac{1}{ℓ} - \frac{1}{ℓ + 1} + \frac{1}{ℓ + 2} + \frac{1}{ℓ + 3} + \dots) d ℓ = \lim_{N \to \infty} [\ln (ℓ + 1) \ln (ℓ + 2) + \ln (ℓ + 3) + \dots)]}_{0}^{N}

\lim_{N \to \infty} [\ln (N) - \ln (N + 1) (N + 2) (N + 3) \dots)] = \lim_{N \to \infty} [\ln (N) - \ln (N!)]

\lim_{N \to \infty} [\ln (N) - (N \ln (N) - N + \frac{1}{2} \ln (2 π N) + O (\frac{1}{N}))] = \infty

diverges

Facebook Tweet Pin