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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}<=nthat 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;
Method of cylinders or method of shells The formula for the area in all of the cases will be, A = 2 ∏ ( radius ) (heig
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In this theorem we identify that for a specified differential equation a set of fundamental solutions will exist. Consider the differential equation y′′ + p (t ) y′ + q (t
find the greater value of a and b so that the following even numbers are divisible by both 3 and 5 : 2ab2a
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The augmented matrix from a system of linear equations has the following reduced row-echelon form (a) How many equations are there in the system? (b) How many variab
X^2 – y^2 – 2y - 1
Bonnie has twice as many cousins as Robert. George has 5 cousins, which is 11 less than Bonnie has. How many cousins does Robert have? Work backwards to find the solution. Geor
Solve for x: 4 log x = log (15 x 2 + 16) Solution: x 4 - 15 x 2 - 16 = 0 (x 2 + 1)(x 2 - 16) = 0 x = ± 4 But log x is
Horizontal asymptotes : Such as we can have vertical asymptotes defined in terms of limits we can also have horizontal asymptotes explained in terms of limits. Definition
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