Failure of Abel's and Dirichlet's test?

Question

Do Abel's test and Dirichlet's test fail in integrals like $\int_0^\infty x^{1/2}e^{-x}$ since $x^{1/2}$ is not convergent in the interval? If I am wrong please correct me !

Answer 1

Abel's test states that if

• $f$ is continuous at $[a,+\infty)$

• $f$ positive and decreasing.

• $\exists M\geq 0\;:\;\forall x\in [a,+\infty)$

$$ \;\;|\int_a^x g(t)dt|\leq M$$

then $\int_a^{+\infty}f(t)g(t)dt$ converges.

for example $\int_1^{+\infty}\sin(t)e^{-t}dt$ is convergent but

$\int_1^{+\infty}\sin(t)dt$ is not.