lim n tends to infintiy ( {x} + {2x} + {3x}..... +{nx}/ n2(to the square) )where {X} denotes the fractional part of x?

Since    0≤{x}<1

          0≤{2x}<1

          0≤{3x}<1

          0≤{4x}<1

          ..........

          0≤{nx}<1

adding all these equation we get,

      0≤{x} + {2x} + {3x} + {4x} + ........ + {nx}<n

implies 0≤({x} + {2x} + {3x} + {4x} + ......... + {nx})/(n^2) < (1/n)

also lim 0 = lim (1/n) = 0 as n tends towards infinity,

therefore by sandwhich theorem,

lim ({x} + {2x} + {3x} + {4x} + ..... + {nx})/ (n^2) = 0 as n tends to infinity.

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;