Compute the Sum

[Phlembac Adib Hasan]

Let $a_0,a_1,a_2,...,a_{2011}$ is a series defined as \[a_n=\frac{n}{2011}\; \; for\; \; n\ge 0\]
Compute the sum \[\sum_{k=0}^{2011}\; \frac{a_k^3}{1-3a_k+3a_k^2}\]
Without using calculator or computer.


(It's one of the coolest APMO problems.Actually I needed only $5$ minutes to solve.So I think it will be a easy problem for Higher Secondary.)

[Phlembac Adib Hasan]

No reply!!! Is it so easy or hard that nobody tries?Well I'm giving a hint:

Use gaussian pairing tool.Also notice \[a^3-3a^2b+3ab^2=(a-b)^3+b^3\]

[*Mahi*]

I think the "too easy" part is true

[zadid xcalibured]

pairing yeids an elegant solution.

[Phlembac Adib Hasan]

I think the "too easy" part is true

I agree.