How can I prove that $\int_{0}^{\infty}\frac{\sin(x)}{x^{\lambda}}$ converges for $\lambda \in (0,2)$?

Asked Nov 04 '19 at 10:00

Active Nov 25 '19 at 04:57

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How can I prove that $\int_{0}^{\infty}\frac{\sin(x)}{x^{\lambda}}$ converges for $\lambda \in (0,2)$?

I tried substituting $\sin(x)$ trough its series represantation in order to get rid of the $x^{\lambda}$ in the denominator when I realised that there the series represantation encompasses $(-1)^k$ and $!$.

How can I go about solving this?

edited Nov 25 '19 at 04:57

Harry Peter

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asked Nov 04 '19 at 10:00

Analysis

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Possible duplicate of Prove that $\int_0^{\infty} \frac{\sin x}{x^p}, dx$ converges for $0<p<2$ – Martin R Nov 04 '19 at 10:16
Also: https://math.stackexchange.com/q/2665616, https://math.stackexchange.com/q/390809. – Martin R Nov 04 '19 at 10:17

How can I prove that $\int_{0}^{\infty}\frac{\sin(x)}{x^{\lambda}}$ converges for $\lambda \in (0,2)$?

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