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Question Number 143755 by Ar Brandon last updated on 18/Jun/21

Study the convergence with respect to  α and β the improper integral below;                   ∫_0 ^∞ (dx/(x^α (lnx)^β ))

$$\mathrm{Study}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to} \\ $$$$\alpha\:\mathrm{and}\:\beta\:\mathrm{the}\:\mathrm{improper}\:\mathrm{integral}\:\mathrm{below}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{dx}}{\mathrm{x}^{\alpha} \left(\mathrm{lnx}\right)^{\beta} } \\ $$

Answered by mathmax by abdo last updated on 18/Jun/21

a lots of case see bertrand integral...

$$\mathrm{a}\:\mathrm{lots}\:\mathrm{of}\:\mathrm{case}\:\mathrm{see}\:\mathrm{bertrand}\:\mathrm{integral}... \\ $$

Commented by Ar Brandon last updated on 18/Jun/21

I′m aware it′s Bertrand′s integral. But can′t find  any reliable link. Can you recommend me one ?

$$\mathrm{I}'\mathrm{m}\:\mathrm{aware}\:\mathrm{it}'\mathrm{s}\:\mathrm{Bertrand}'\mathrm{s}\:\mathrm{integral}.\:\mathrm{But}\:\mathrm{can}'\mathrm{t}\:\mathrm{find} \\ $$$$\mathrm{any}\:\mathrm{reliable}\:\mathrm{link}.\:\mathrm{Can}\:\mathrm{you}\:\mathrm{recommend}\:\mathrm{me}\:\mathrm{one}\:? \\ $$

Commented by mathmax by abdo last updated on 20/Jun/21

take a look at google ...

$$\mathrm{take}\:\mathrm{a}\:\mathrm{look}\:\mathrm{at}\:\mathrm{google}\:... \\ $$

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