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calculer-lim-n-oo-f-n-x-f-n-x-0-oo-ne-x-1-nx-dx-x-0-oo-

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Question Number 207634 by SANOGO last updated on 21/May/24
calculer lim n→+oo f_n (x) f_n (x)=∫_0^ ^(+oo) ((ne^(−x) )/(1+nx))dx /x∈[0+oo[
calculerlimn→+oofn(x)fn(x)=∫0+oone−x1+nxdx/x∈[0+oo[
Answered by Berbere last updated on 21/May/24
i Think is U_n =∫_0 ^∞ ((ne^(−x) )/(1+nx))dx=^(IBP) [ln(1+nx)e^(−x) ]_0 ^∞ +∫_0 ^∞ ln(1+nx)e^(−x) dx =∫_0 ^1 ln(1+nx)e^(−x) +∫_1 ^∞ ln(1+nx)e^(−x) dx ∫_0 ^1 ln(1+nx)e^(−x) dx>0⇒U_n >∫_1 ^∞ ln(1+nx)e^(−x) dx ∀x≥1;ln(1+nx)≥ln(1+n)⇒U_n ≥∫_1 ^∞ ln(1+n)e^(−x) dx =ln(1+n)[−e^(−x) ]=((ln(1+n))/e) U_n >((ln(1+n))/e)⇒U_n →+∞
iThinkisUn=∫0∞ne−x1+nxdx=IBP[ln(1+nx)e−x]0∞+∫0∞ln(1+nx)e−xdx=∫01ln(1+nx)e−x+∫1∞ln(1+nx)e−xdx∫01ln(1+nx)e−xdx>0⇒Un>∫1∞ln(1+nx)e−xdx∀x⩾1;ln(1+nx)⩾ln(1+n)⇒Un⩾∫1∞ln(1+n)e−xdx=ln(1+n)[−e−x]=ln(1+n)eUn>ln(1+n)e⇒Un→+∞
Commented by SANOGO last updated on 21/May/24
merci beaucoup
mercibeaucoup

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