Chapter 11 - Infinite Sequences and Series - 11.3 The Integral Test and Estimates of Sums - 11.3 Exercises - Page 767: 45

$b\lt \frac{1}{e}$ (with $b\gt 0$).

$\Sigma_{n=1}^{\infty}b^{ln(n)}=\Sigma_{n=1}^{\infty}n^{lnb}=\Sigma_{n=1}^{\infty}\frac{1}{n^{-lnb}}$ This is a p-series with $p=-lnb$ So if $p=-lnb\gt 1$, then the series will converge. $$-lnb\gt 1$$$$lnb\lt -1$$$$b\lt e^{-1}$$$$b\lt \frac{1}{e}$$ Hence, $b\lt \frac{1}{e}$ (with $b\gt 0$).