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Grade: 12th pass
        
Please solve with steps and give a neat solution as soon as possible please
4 months ago

Answers : (1)

Arun
23020 Points
							
 

x] can be written as X+{x}    --> {x} is fractional part .. between 0 and 1

 

therefore , 

lim n→∞  [x]+[2x]+[3x]+.........+[nx]


                                  n2 

is = 

lim n→∞  x+2x+3x+.........+nx + {x}+{2x} ...+{nx}


                                  n2 

 

= x(1+2+3+..+n)   +   {x}+...{nx}


                           n^2

 

 

=x(n(n+1))/2n^2 +  ({x}+...{nx})/n^2

 

Since {} is only b/w 0 and 1  , the second operand becomes 0 as n tends to ∞

 

on solving the first part , u get (n2 x +nx)/2n2 = x/2 +x/2n

 

x/2n becomes 0 as x tends to infinity.

 

therefore the answer is x/2.

4 months ago
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