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For-all-n-N-f-n-x-nx-2n-1-x-0-2n-1-n-1-x-2n-1-n-2-then-for-n-1-2-f-n-x-dx-convergences-to-

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Question Number 55638 by gunawan last updated on 01/Mar/19
For all n ∈ N f_n (x)= { ((((nx)/(2n−1)), x ∈ [0, ((2n−1)/n)])),((1 , x ∈[((2n−1)/n), 2])) :} then for n→∞ ∫_1 ^2 f_n (x) dx convergences to..
Foralln∈Nfn(x)={nx2n−1,x∈[0,2n−1n]1,x∈[2n−1n,2]thenforn→∞∫12fn(x)dxconvergencesto..
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19
∫_1 ^((2n−1)/n) ((nx)/(2n−1))dx+∫_((2n−1)/n) ^2 1×dx (n/(2n−1))×∣^ (x^2 /2)∣_1 ^((2n−1)/n) +∣x∣_((2n−1)/n) ^2 =(n/(2(2n−1)))×{(((2n−1)/n))^2 −1}+2−(((2n−1)/n)) =(1/2)×((2n−1)/n)−(n/(2(2n−1)))+2−((2n−1)/n) =2−(((2n−1))/(2n))−(n/(2(2n−1))) =2−((2−(1/n))/2)−((n/n)/(2(((2n)/n)−(1/n)))) =2−((2−(1/n))/2)−(1/(2(2−(1/n)))) when n→∞ =2−((2−0)/2)−(1/(2(2−0))) =2−1−(1/4) 1−(1/4)=(3/4)
∫12n−1nnx2n−1dx+∫2n−1n21×dxn2n−1×∣x22∣12n−1n+∣x∣2n−1n2=n2(2n−1)×{(2n−1n)2−1}+2−(2n−1n)=12×2n−1n−n2(2n−1)+2−2n−1n=2−(2n−1)2n−n2(2n−1)=2−2−1n2−nn2(2nn−1n)=2−2−1n2−12(2−1n)whenn→∞=2−2−02−12(2−0)=2−1−141−14=34

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