I was asked to find the limit of the summation of a series which consisted of variables inside the greatest integer function. My textbook says the following :-
For any integer $k$ , we will have
$kx-1 < [kx] ≤ kx$
I was able to verify this for small numbers like $3,4$ etc.
Also for the next sum says
For any integer $k$ , we will have
$k^2x-1 <[k^2x] ≤ k^2x +1$
now i am confused as to why the additional +1 appears on the upper limit in case of $[k^2x]$ but not $[kx]$. If , for instance if we're asked to find the range of $[k^3x]$ how would I derive it ? Thanks for your help
note $[.]$ represents the greatest integer function