Cauchy Integral Test/Motivation

Cauchy Integral Test: Motivation

It follows from the Cauchy Integral Test that if $f$ is continuous, positive and decreasing on $\hointr 1 \infty$, then the series $\ds \sum_{k \mathop = 1}^\infty \map f k$ and the improper integral $\ds \int_1^{\mathop \to +\infty} \map f x \rd x$ either both converge or both diverge.

So this theorem provides a test for the convergence of both a series and an improper integral.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.32$