Proposition 2

Let $(u_n)$ and $(v_n)$ be two series with positive terms: 1. If the series $(v_n)$ converges then the relation $u_n \ll v_n$ (resp. $u_n \sim v_n$) implies $\sum_{p=n}^{\infty} u_p \ll \sum_{p=n}^{\infty} v_p$ (resp. $\sum_{p=n}^{\infty} u_p \sim \sum_{p=n}^{\infty} v_p$). 2. If $\sum_{n=1}^{\infty} v_n = +\infty$ then the relation $u_n \ll v_n$ (resp. $u_n \sim v_n$) implies $\sum_{p=1}^{n} u_p \ll \sum_{p=1}^{n} v_p$ (resp. $\sum_{p=1}^{n} u_p \sim \sum_{p=1}^{n} v_p$).

One obtains convenient convergence criteria by taking for the comparison series $(v_n)$ in prop. 1 a series whose terms are of the form $v_n = f(n)$, where $f$ is a function $\geq 0$ defined for every real number $x > x_0$ and decreasing on the interval $[x_0, +\infty[$; indeed: