I know this following formulation of Abel's test for improper integrals:
Let $f,g:[a,\infty)\to \mathbb{R}$ be continuous functions, where
- $\int_a^\infty f(t)dt$ converges.
- $g$ is monotone decreasing, bounded and continuously differentiable.
Then $\int_a^\infty f(t)g(t)dt$ converges.
The proof I know uses integration by parts, so I am not sure if one can strengthen it, but I'm looking for counter-examples of stronger versions. More specifically, when $f$ is only Riemann integrable on any compact interval, but I currently can't think of any.
Best Answer
Weak sufficient conditions are that $\int_a^\infty f(x) \, dx$ converges and $g$ is bounded and monotone (decreasing or increasing). By the second mean value theorem for integrals, there exists $\xi \in [c_1,c_2]$ such that
$$\int_{c_1}^{c_2} f(x) g(x) \, dx = g(c_1) \int_{c_1}^\xi f(x) \, dx + g(c_2)\int_\xi^{c_2} f(x) \, dx,$$
Since $g$ is bounded with $|g(x)| \leqslant M$, we have
$$\left|\int_{c_1}^{c_2} f(x) g(x) \, dx\right|\leqslant |g(c_1)| \left|\int_{c_1}^\xi f(x) \, dx\right| + |g(c_2)|\left|\int_\xi^{c_2} f(x) \, dx\right|\\ \leqslant M \left|\int_{c_1}^\xi f(x) \, dx\right| + M\left|\int_\xi^{c_2} f(x) \, dx\right|$$
For every $\epsilon > 0$ there exists $C > a$ such that for all $c_2 \geqslant \xi \geqslant c_1 > C$,
$$ \left|\int_{c_1}^\xi f(x) \, dx\right|, \,\,\left|\int_\xi^{c_2} f(x) \, dx\right| < \frac{\epsilon}{2M},$$
which implies
$$\left|\int_{c_1}^{c_2} f(x) g(x) \, dx\right| < \epsilon$$
Thus, $\int_a^\infty f(x) g(x) \, dx$ converges by the Cauchy criterion.
A counterexample where $\int_a^\infty f(x) \, dx$ fails to converge -- while other conditions are met -- has been given in comments.
Also, the convergence of $\int_a^\infty f(x) \, dx$ is not a necessary condition. An example is $a=1$, $f(x) = g(x) = 1/x$, where
$$\int_1^\infty f(x) \, dx = \infty, \,\,\int_1^\infty f(x)g(x) \, dx = 1$$