Chapter 11 - Infinite Sequences and Series - 11.3 The Integral Test and Estimates of Sums - 11.3 Exercises - Page 766: 40

$N\geq e^{100}$

$\int_{N}^{\infty}\frac{1}{x(lnx)^{2}}dx\leq 0.01$ $[-\frac{1}{lnx}]_{N}^{\infty}\leq 0.01$ $-\frac{1}{ln\infty}-(-\frac{1}{lnN})\leq 0.01$ $\frac{1}{lnN}\leq \frac{1}{100}$ $lnN\geq 100$ Hence, $N\geq e^{100}$