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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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    What measure do you use to say that his question almost always have exclamations? :P2012-10-27

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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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    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.2013-07-23