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Question Number 27601 by abdo imad last updated on 10/Jan/18

f fonction numerical increasing on ]0,1] and  ∫_0 ^1 f(t)dt converges prove that  lim_(n−>∝)   (1/n) Σ_(k=1) ^n  f((k/n))  = ∫_0 ^1 f(t)dt  .

$$\left.{f}\left.\:{fonction}\:{numerical}\:{increasing}\:{on}\:\right]\mathrm{0},\mathrm{1}\right]\:{and} \\ $$ $$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({t}\right){dt}\:{converges}\:{prove}\:{that}\:\:{lim}_{{n}−>\propto} \:\:\frac{\mathrm{1}}{{n}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}}\right) \\ $$ $$=\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({t}\right){dt}\:\:. \\ $$

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