Divergence of an integral related to a Riemann integral $\int_{1}^{\infty}\dfrac{1}{x}dx$

calculus improper-integrals riemann-integration

Each integral needs to converge separately. As a definition, $$\int_{-\infty}^\infty f(x)\mathrm dx=\lim_{A\to -\infty}\lim_{B\to \infty} \int_A^B f(x)\mathrm dx$$ Since neither of these limits exist for our $f$, the integral is divergent.

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