= Dirichlet test for improper integrals
{c}
If $F(t)=\int_a^t f(x)\,dx$ is bounded and $g$ is a nonnegative differentiable decreasing <function> tending to zero, then $\int_a^\infty f(x)g(x)\,dx$ converges as an <improper integral>. <Integration by parts> bounds the tail $\int_A^Bfg$ by a constant multiple of $g(A)$, because $F$ is bounded and $\int_A^B|g'|=g(A)-g(B)$. In particular $\int_1^\infty x^{-\alpha}\sin x\,dx$ converges for every $\alpha>0$, even though <absolute integrability> of that tail requires $\alpha>1$.
Back to article page
