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How can I prove that the following (Lebesgue) integral is $+\infty$ for every $t\neq0$ and $a>0$? I tried to break it so that I don't have absolute value, first in 3 integrals, then in 2 by changing the variable and calculate only one of the integrals. Every time I get either $\infty \cdot0$ or $\infty-\infty$.

$$\int_{|x|\geq1}e^{tx}\frac{a}{2|x|^{a+1}}dλ(x), $$

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  • $\begingroup$ Suffices to show the integral over $x > 1$ diverges, the notice $e^{tx} \geq c (tx)^{a+2}$ for some $c$ (you can in fact know $c$ by Taylor's series). $\endgroup$
    – William M.
    Aug 19, 2021 at 16:45

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This helped

Suffices to show the integral over $x> 1$ diverges, using the notice $e^{tx}≥c(tx)^{a+2}$ for some $c$ (you can in fact know $c$ by Taylor's series)

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