Sequence: integral of integer part

[MM]

Define the sequence [tex]\{a_n\}_{n\ge1}[/tex] for which [tex]a_n=\int_{-2n}^{2n} \lfloor x\rfloor x \ dx[/tex] where [tex]\lfloor x\rfloor[/tex] denotes the largest integer which doesn't exceed [tex]n[/tex]. Express [tex]a_n[/tex] in terms of [tex]n[/tex].

[MM]

This may be useful.

Spoiler: show

Prove that [tex]\int_{-2n}^{0} \lfloor x\rfloor x \ dx-\int_{0}^{2n} \lfloor x\rfloor x \ dx=2n^2[/tex].

[Guest]

where [tex]\lfloor x\rfloor[/tex] denotes the largest integer which doesn't exceed [tex]n[/tex].

That means [tex]\lfloor x\rfloor=n-1[/tex]. I don't think so.

[MM]

Excuse me. I meant "which doesn't exceed [tex]x[/tex]".