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

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

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

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## Comments

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

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Precise! So 0 is correct answer.

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

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

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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.

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