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# Weird!!!

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.

Note by Jaydee Lucero
3 years, 9 months ago

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Note 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. · 3 years, 9 months ago

Precise! So 0 is correct answer. · 3 years, 9 months ago

To evaluate the sum, note that terms of opposite signs cancel, and the only term left is $$0^3 = 0.$$ · 3 years, 9 months ago

now I see... thanks... :D · 3 years, 9 months ago