$ p $级数测试和应用程序

Overview of the $p$-series test and its uses

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.