Question Number 101582 by Rohit@Thakur last updated on 03/Jul/20

lim_(n→∞)   (1/n^2 ) Σ_(r=1) ^n  r e^(r/n)  =

Answered by mathmax by abdo last updated on 03/Jul/20

A_n =(1/n^2 )Σ_(k=1) ^(n ) k e^(k/n)  ⇒A_n =(1/n) Σ_(k=1) ^n  (k/n)e^(k/n)   so A_n is a Rieman sum and  lim_(n→+∞)  A_n =∫_0 ^1  xe^x  dx =[x e^x ]_0 ^1  −∫_0 ^1  e^x  dx  =e−(e−1) =1

Answered by Dwaipayan Shikari last updated on 03/Jul/20

(1/n)lim_(n→∞) Σ_(r=1) ^n (r/n)e^(r/n) =∫_0 ^1 xe^x dx=[xe^x ]_0 ^1 −[e^x ]_0 ^1 =1

Commented byRohit@Thakur last updated on 04/Jul/20

Thank you sir