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Suppose that $f\in L^1(0,+\infty)$ is a monotone decreasing, positive function. Prove that $$\lim_{x \to +\infty}x(\log x)\cdot f(x)=0.$$

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    xln(x)f(x) $\qquad$ or $\qquad$ xln(xf(x))??2012-12-02
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    $f(x) x \ln x $2012-12-02
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    The problem is to see whether there is a sequence $\{t_k\}$ of real numbers, increasing to $+\infty$ and such that the series $\sum_k\frac{t_k-t_{k-1}}{t_k\log t_k}$ is convergent.2012-12-02
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    @DavideGiraudo Good analysis. There is.2012-12-02
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    Yes, for example $t_k=2^{k^2}$.2012-12-03
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    OP: To modify drastically the question after some answers are posted is contrary to the policy of the site (and to politeness, I should add). Please do not do this. If you have a question different from the one posted, then post another question. (I reverted your question to its previous version.)2012-12-05
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    I am very sorry. I tried to post another question but the system told me that it is the same as the above one. I would like to post now the question with the additional condition, f to be convex. Could anybody help me to post the new question?2012-12-07

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