Statement of Abel's test and Dirichlet's test for unbounded intervals

Question

I know both Abel's and Dirichlet's test for convergence of some integrals when the range of the integral is of the form [a, b), where b can be $\infty$. How can i use these test for functions defined on intervals like (-$\infty$, b]? Please explain me the method. I research all the internet for an article, but only found for intervals [a, b). I saw the demonstration and it seems to work for (-$\infty$, b] but idk... please explain me step by step.

Answer 1

In this case we have an integrand $f: (-\infty,b] \to \mathbb{R}$ which can be expressed as the product of functions $f = gh$ and such that $f$ is Riemann integrable on $[a,b]$ for all $-\infty < a < b$. These tests provide sufficient conditions on $g$ and $h$ for convergence of the improper integral

$$\int_{-\infty}^b g(x) h(x) \, dx = \lim_{a \to -\infty}\int_a^b g(x) h(x) \, dx$$

For Dirichlet's test, the conditions are (1) $\left|\int_a^b g(x) \, dx\right|$ is uniformly bounded for all $a < b$, and (2) $h(x) \to 0$ (monotonically) as $x \to -\infty$.

The only difference from the conditions for the convergence of an improper integral over $[a, +\infty)$ are that (1) the integral of $g$ is bounded for all $a < b$ with $b$ fixed, rather than for all $b > a$ with $a$ fixed, and that (2) $h(x) \to 0$ monotonically as $x \to -\infty$, rather than as $x \to +\infty$.

The proof is virtually identical as it involves showing that the Cauchy criterion holds, that is for all $\epsilon > 0$ there exists $C(\epsilon)$ such that

$$\left| \int_{c_1}^{c_2} g(x) h(x) \, dx\right|< \epsilon,$$

for all $-\infty < c_1 < c_2 < C(\epsilon)< b$ (rather than $a < C(\epsilon) < c_1 < c_2 < +\infty$).

To do this we apply the second mean value theorem for integrals which implies that if $h$ is monotone then there exists $\xi \in (c_1,c_2)$ such that

$$\left|\int_{c_1}^{c_2}g(x) h(x) \, dx\right| = \left|h(c_1)\int_{c_1}^{\xi}g(x) \, dx + h(c_2)\int_{\xi}^{c_2}g(x) \, dx \right| \\ \leqslant |h(c_1)|\left|\int_{c_1}^{\xi}g(x) \, dx\right| + |h(c_2)|\left|\int_{\xi}^{c_2}g(x) \, dx\right|$$

The sign of the integration limits are irrelevant. What matters is that conditions (1) and (2) are met and, in particular, that $h(x) \to 0$ as $x \to -\infty$.

Similar reasoning applies in showing Abel's test holds as well.