Expii

# 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.