$f:[a,{\infty}[\to\mathbb{R}\ $ is bounded and suppose that $f$ is integrable on each interval of the form $[a,b[$. Prove that $$\int_0^\infty \frac { f(x) }{ x^p } \ \, dx$$ converges absolutely for $p>1$. Conclude that, if $x^pf(x)$is bounded for some $p>1$, then $$\int_a^\infty f\ \, dx$$ converges absolutely.

  • $\begingroup$ Did you mean $[a,b]$ $\endgroup$ – Justin Oct 13 '13 at 2:03
  • $\begingroup$ That might not be integrable. Consider $f(x)=1$. $\endgroup$ – Jack M Oct 13 '13 at 2:24

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