I have got stuck with a question. Please help me. Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$. Thank You.
2026-04-28 08:19:06.1777364346
Absolute Convergence of a Function
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Since $$\lim_{x\to0}\left(\frac{\sin x}{x}\right)^p=1$$ then the integral $$\int_0^1\left|\frac{\sin x}{x}\right|^p dx\quad \forall p$$ is convergent, moreover we have $$\left|\frac{\sin x}{x}\right|^p\leq\frac{1}{x^p} $$ and since the integral $$\int_1^\infty \frac{dx}{x^p}\quad\text{is convergent if}\quad p>1$$ then we can conclude that $x\mapsto\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$.