Proposition 3 (Cauchy-Maclaurin criterion)

If $f$ is a function $\geq 0$ and decreasing on $[x_0, +\infty[$, then the series with general term $v_n = f(n)$ converges if and only if the integral $\int_{x_0}^{+\infty} f(t)\,dt$ converges.

To prove this it is sufficient to note that if $v$ is the step function associated with the series $(v_n)$ then $v(x+1) \leq f(x) \leq v(x)$ for every $x \geq x_0$ since $f$ is decreasing; the proposition is thus a consequence of the comparison principle (II, p. 66, prop. 3).

Since the functions which feature in the logarithmic convergence criteria for integrals (V, p. 229, prop. 2, 3 and 4) are decreasing on an interval $[x_0, +\infty[$, applying prop. 1 and 3 of V, p. 237, gives the following criteria: