p-Series Test and Applications - Expii

The integral test, applied to the function \(f(x) = x^{-p} = 1/x^p\), is called the \(p\)-series test: the \(p\)-series \(1^{-p} + 2^{-p} + 3^{-p} + \dots\) is convergent if \(p>1\) and divergent otherwise, if \(p\leq 1\). (In fact, when \(p>1\), the Riemann zeta function \(\zeta\) is defined by this series: \(\zeta(p) := 1^{-p} + 2^{-p} + 3^{-p} + \dots\).) The \(p\)-series test is often useful when applying series comparison tests.