Question Number 101791 by Dwaipayan Shikari last updated on 04/Jul/20

(1/n^(3  ) )lim_(n→∞) (ne^(−((1/n))^2 ) +2ne^(−((2/n))^2 ) +....∞)

Answered by Ar Brandon last updated on 04/Jul/20

lim_(n→∞) (1/n^3 ){ne^(−((1/n))^2 ) +2ne^(−((2/n))^2 ) +∙∙∙+n∙ne^(−((n/n))^2 ) }  A_n =lim_(n→∞) (1/n^3 )Σ_(k=1) ^n {kne_ ^(−((k/n))^2 ) }=lim_(n→∞) (1/n)Σ_(k=1) ^n {(k/n)e^(−((k/n))^2 ) }        =∫_0 ^1 xe^(−x^2 ) dx=(1/(−2))[e^(−x^2 ) ]_0 ^1 =(1/2){1−(1/e)}

Commented byDwaipayan Shikari last updated on 04/Jul/20

You are right sir . It will be (1/n^3 )

Answered by mathmax by abdo last updated on 04/Jul/20

A_n =(1/n^3 )Σ_(k=1) ^n  nk e^(−(k^2 /(n^2  )))  ⇒ A_n =(1/n) Σ_(k=1) ^n  (k/n)e^(−((k/n))^2 )  ⇒  lim_(n→+∞)  A_n =∫_0 ^1  xe^(−x^2 ) dx =[−(1/2)e^(−x^2 ) ]_0 ^1  =(1/2) −(1/2)e^(−1)  =(1/2)(1−(1/e))  =((e−1)/(2e))

Commented byDwaipayan Shikari last updated on 04/Jul/20

Thanking you

Commented bymathmax by abdo last updated on 05/Jul/20

you are welcome