Question

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

Answer 1

`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`