What is the value of #lim_(n->oo)sum_(r=1)^n({r^2x})/n^3# , if #{.}# denotes the fractional part of the function?

1 Answer
Jan 21, 2018

#0#

Explanation:

#{x}=x-floor(x)# then

#sum_(k=1)^n {{k^2 x}}/n^3 = sum_(k=1)^n (k^2 x)/n^3-sum_(k=1)^n floor(k^2x)/n^3#

Here

#S_1(n)=sum_(k=1)^n (k^2 x)/n^3 = sum_(k=1)^n ((k/n)^2 x)1/n#

and calling #xi = k/n, d xi = 1/n# we can use the Riemann equivalent instead.

#lim_(n->oo)S_1(n) = int_0^1 xi^2 x d xi = x/3#

and knowing that

#lim_(n->oo) floor(n x)/n = x# we have

#S_2(n) = sum_(k=1)^n floor(n^2(k/n)^2 x)/n^2 1/n# so

#lim_(n->oo)S_2(n) = int_0^1 xi^2 x d xi = x/3# and finally

#sum_(k=1)^n {{k^2 x}}/n^3 =lim_(n->oo)(S_1(n)-S_2(n)) = 0#