lim n tends to infintiy ( {x} + {2x} + {3x}..... +{nx}/ n2(to the square) )where {X} denotes the fractional part of x?
Ans) all no.s are positive or 0. so limit is either positive or 0.........(1)
now {x}<=1;{2x}<=1;......
{x}+{2x}+....{nx}<=n
that implies lim n tends to infinity{x}+{2x}+....{nx}/n^2 <= lmit n tends to infinity n/n^2;
that means lim n tends to infinity{x}+{2x}+....{nx}/n^2 <= lmit n tends to infinity 1/n i.e. 0........(2)
from (1) and (2);
required limit=0;