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let-f-n-x-1-1-x-n-1-1-n-ddfined-on-0-1-1-prove-that-f-n-cs-f-n-2-calculate-I-n-0-1-f-n-x-dx-

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Question Number 40150 by maxmathsup by imad last updated on 16/Jul/18
let f_n (x) =(1/((1+x^n )^(1+(1/n)) )) ddfined on [0,1] 1) prove that f_n →^(cs) f (n→+∞) 2) calculate I_n = ∫_0 ^1 f_n (x)dx
letfn(x)=1(1+xn)1+1nddfinedon[0,1]1)provethatfn→csf(n→+∞)2)calculateIn=∫01fn(x)dx
Commented by maxmathsup by imad last updated on 21/Jul/18
1) we have f_n (x) =(1+x^n )^(−(1+(1/n))) =e^(−(1+(1/n))ln(1+x^n )) but ln(1+x^n )∼ x^n (n→+∞) ⇒−(1+(1/n))ln(1+x^n ) ∼−x^n −(x^n /n) ⇒ f_n (x) ∼ e^(−x^n −(x^n /n)) →^(cs) f(x)=1 (n→+∞)
1)wehavefn(x)=(1+xn)−(1+1n)=e−(1+1n)ln(1+xn)butln(1+xn)∼xn(n→+∞)⇒−(1+1n)ln(1+xn)∼−xn−xnn⇒fn(x)∼e−xn−xnn→csf(x)=1(n→+∞)

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