let-f-n-x-1-1-x-n-1-1-n-defined-on-0-1-1-prove-that-f-n-cs-to-a-function-f-on-0-1-2-calculate-I-n-0-1-f-n-x-dx- Tinku Tara June 3, 2023 Integration 0 Comments FacebookTweetPin Question Number 66344 by mathmax by abdo last updated on 12/Aug/19 letfn(x)=1(1+xn)1+1ndefinedon[0,1]1)provethatfn→cstoafunctionfon[0,1]2)calculateIn=∫01fn(x)dx Commented by mathmax by abdo last updated on 19/Aug/19 1)onafn(0)=1andfn(1)=121+1n→12(n→+∞)andfor0<x<1fn(x)=(1+xn)−(1+1n)=e−(1+1n)ln(1+xn)wehavexn→0⇒ln(1+xn)∼xn⇒−(1+1n)ln(1+xn)∼−(1+1n)xn⇒fn(x)∼e−(1+1n)xn⇒fncs→1on]0,1[ Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: if-the-equations-of-the-sides-of-the-triangle-are-7x-y-10-0-x-2y-5-0-and-x-y-2-0-find-the-orhocentre-of-the-triangle-Next Next post: let-I-n-0-1-x-2n-1-ln-x-x-2-1-dx-1-prove-the-existence-of-I-n-2-calculate-I-n-1-I-n-3-prove-thst-x-0-1-0-lt-xlnx-x-2-1-lt-1-2-4-find-lim-n-I-n- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.
![let f_n (x)=(1/((1+x^n )^(1+(1/n)) )) defined on [0,1] 1)prove that f_n →^(cs) to a function f on[0,1] 2) calculate I_n =∫_0 ^1 f_n (x)dx](https://www.tinkutara.com/question/Q66344.png)
![1) ona f_n (0) =1 and f_n (1) =(1/2^(1+(1/n)) ) →(1/2) (n→+∞) and for 0<x<1 f_n (x) =(1+x^n )^(−(1+(1/n))) =e^(−(1+(1/n))ln(1+x^n )) we have x^n →0 ⇒ ln(1+x^n )∼x^n ⇒−(1+(1/n))ln(1+x^n )∼−(1+(1/n))x^n ⇒ f_n (x)∼e^(−(1+(1/n))x^(n ) ) ⇒ f_n ^( cs) → 1 on]0,1[](https://www.tinkutara.com/question/Q66811.png)