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The comparison test decides convergence or divergence by bounding the target integrand with a simpler nonnegative function. If $0 \le g(x) \le f(x)$ on the tail, convergence of the larger integral $\int_a^\infty f(x)\,dx$ forces convergence of the smaller integral $\int_a^\infty g(x)\,dx$.

Conditions: The functions $f(x)$ and $g(x)$ are nonnegative on the tail interval $[a, \infty)$.; The inequality $0 \le g(x) \le f(x)$ holds for all $x \ge a$.; Finite-interval integrability is assumed for the functions involved.

The comparison test decides convergence or divergence by bounding the target integrand with a simpler nonnegative function. If $0 \le g(x) \le f(x)$ on the tail, convergence of the larger integral $\int_a^\infty f(x)\,dx$ forces convergence of the smaller integral $\int_a^\infty g(x)\,dx$.

Conditions: The functions $f(x)$ and $g(x)$ are nonnegative on the tail interval $[a, \infty)$.; The inequality $0 \le g(x) \le f(x)$ holds for all $x \ge a$.; Finite-interval integrability is assumed for the functions involved.