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Find the value of x=20132013xxxsgn(x)\sum\limits_{x=-2013}^{2013} x \cdot |x| \cdot \lfloor x \rfloor \cdot \text {sgn} (x) .

Here, x\lfloor x \rfloor denotes the greatest integer function or floor function, and sgn(x)\text {sgn} (x) denotes the sign function: 1 if xx is positive, 0 if xx is zero, and -1 if xx is negative.

I have invented this question, and got the answer 0\boxed{0} (correct me if I'm wrong), but I don't know how to solve this without using graphs. Please help. Thanks.

Note by Jaydee Lucero
5 years, 11 months ago

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3 votes

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Note that x=x\lfloor x \rfloor = x as xx is an integer, and xsgn(x)=x|x| \cdot \text{sgn}(x) = x for all xx. So you're basically asking x=20132013x3\displaystyle\sum_{x=-2013}^{2013} x^3, which is now easy to solve.

Ivan Koswara - 5 years, 11 months ago

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Precise! So 0 is correct answer.

Piyushkumar Palan - 5 years, 11 months ago

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To evaluate the sum, note that terms of opposite signs cancel, and the only term left is 03=0.0^3 = 0.

Michael Tang - 5 years, 11 months ago

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now I see... thanks... :D

Jaydee Lucero - 5 years, 11 months ago

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by graph: The above function takes values of x on the X-axis and y=f(x) on the Y axis. this question is basically to plot the graph of f(x)=x^3, here we are only taking the integers values at x-axis and getting correspond y values as integers.So instead of smooth curve we will will be having the points which lies on the curve of f(x)=x^3.

Shubham Namdeo - 5 years, 11 months ago

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