Question Number 101461 by Dwaipayan Shikari last updated on 02/Jul/20

Answered by MAB last updated on 02/Jul/20

lim_(n→∞) Σ_(r=1) ^(4n) ((√n)/((√r)(3(√r)+4(√n))^2 ))=lim_(n→∞) (1/n)Σ_(r=1) ^(4n) (1/((√(r/n))(3(√(r/n))+4)^2 ))  =lim_(N→∞) (1/N)Σ_(r=1) ^N (2/((√(r/N))(6(√(r/N))+4)^2 ))  (4n=N)  =∫_0 ^1 (2/((√x)(6(√x)+4)^2 ))dx   (u=(√x))  =∫_0 ^1 (2/(u(6u+4)^2 ))2udu  =∫_0 ^1 (1/((3u+2)^2 ))du  =[((−1)/3)∙(1/(3u+2))]_0 ^1   =(1/(10))  finally:  lim_(n→∞) Σ_(r=1) ^(4n) ((√n)/((√r)(3(√r)+4(√n))^2 ))=(1/(10))

Commented byDwaipayan Shikari last updated on 02/Jul/20

Thanking you

Commented byMAB last updated on 02/Jul/20

you are welcome