Chapter 11 - Infinite Sequences and Series - 11.3 The Integral Test and Estimates of Sums - 11.3 Exercises - Page 766: 29

Converges by p-series for $p\gt 1$

For $p\gt 1$ $\int_{2}^{\infty}\frac{1}{x(lnx)^{p}}dx=[\frac{1}{(1-p)(lnx)^{p-1}}]_{2}^{\infty}$ $=\frac{1}{(1-p)(ln\infty)^{p-1}}-\frac{1}{(1-p)(ln2)^{p-1}}$ $=\frac{1}{-\infty}-\frac{1}{(1-p)(ln2)^{p-1}}$ $=-\frac{1}{(1-p)(ln2)^{p-1}}$ Converges by p-series for $p\gt 1$ .