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Question Number 55051 by maxmathsup by imad last updated on 16/Feb/19

calculate lim_(n→+∞) ∫_0 ^1 (dx/(1+x+x^2 +...+x^n ))

calculatelimn→+∞∫01dx1+x+x2+...+xn

Commented by maxmathsup by imad last updated on 24/Feb/19

let I_n =∫_0 ^1 (dx/(1+x+x^2 +...+x^n )) ⇒I_n =∫_0 ^1 (dx/((1−x^(n+1) )/(1−x))) =∫_0 ^1 ((1−x)/(1−x^(n+1) )) dx =∫_R ((1−x)/(1−x^(n+1) )) χ_(]0,1[) (x) dx the sequence of function f_n (x)=((1−x)/(1−x^(n+1) )) χ_(]0,1[) (x) converges to f(x)=1−x ⇒ im_(n→+∞) I_n =∫_0 ^1 (1−x)dx =[x−(x^2 /2)]_0 ^1 =1−(1/2) =(1/2)

letIn=∫01dx1+x+x2+...+xn⇒In=∫01dx1−xn+11−x=∫011−x1−xn+1dx=∫R1−x1−xn+1χ]0,1[(x)dxthesequenceoffunctionfn(x)=1−x1−xn+1χ]0,1[(x)convergestof(x)=1−x⇒imn→+∞In=∫01(1−x)dx=[x−x22]01=1−12=12

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