Question Number 101373 by Dwaipayan Shikari last updated on 02/Jul/20

lim_(n→∞ ) Σ_(r=1) ^(4n) ((√n)/((√r)(3(√r)+4(√n))^2 ))

Commented byDwaipayan Shikari last updated on 02/Jul/20

(1/n)lim_(n→∞) Σ_(r=1) ^(4n) ((n/r))^(3/2) .(1/((3+4(√(n/r)))^2 ))=∫_0 ^4 (1/x^(3/2) ).(1/((3+(4/x^(1/2) ))^2 ))dx=∫_0 ^4 ((1/x^(3/2) )/((3+(4/x^(1/2) ))^2 ))dx −(1/2)∫_0 ^4 (((−2)/x^(3/2) )/((3+(4/x^(1/2) ))^2 ))dx=−(1/2)∫_∞ ^5 (dt/t^2 )=(1/2)[(1/t)]_∞ ^5 =(1/(10)) Is it right???????