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Suppose $f'(x)$ exists on $[0,\infty)$, prove or disprove that: the following two integrals $$\int_{0}^{+\infty}\frac{2dx}{f(x)} \ \ \text{and}\ \int_{0}^{+\infty}\frac{dx}{f(x)+f'(x)}$$ have the same convergence.

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Why are your questions almost always exclamations! :-) – commenter Oct 26 '12 at 9:32
Maybe He/she loves factorializing. – Sangchul Lee Oct 26 '12 at 12:08
What measure do you use to say that his question almost always have exclamations? :P – Jean-Sébastien Oct 27 '12 at 0:57
up vote 1 down vote accepted

This is not true. For a counterexample, use $f(x) = e^{-x}$ on $[0,1]$, $f(x) = e^{x}$ on $[2,\infty)$, and interpolate smoothly with a positive function on $[1,2]$. (Obviously here the problem is on $[0,1]$, not at $\infty$.)

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Thank you, I think I got it! Thank you Pro. Lukas Geyer. I know you. I am also interested in Holomorphic Dynamics, especially the linearizable problem near the irrational fixed point of rational maps. – Riemann Oct 27 '12 at 1:13
I cannot understand it. We are assuming that $f'(x)\ge0$ right? How could we use $e^{-x}$? And I do not find this to be a counterexample. Could you tell me what I am missing? Thanks. – awllower Jul 23 '13 at 9:20

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