Question

et (1-1/2^2)(1-1/3^2)...(1-1/2010^2)(1-1/2011^2) = x/2*2011. what is the value of x?

Answer 1

\[\left(\left(1-\frac{1}{2^2}\right) \left(1-\frac{1}{3^2}\right)\text{...}\right) \left(1-\frac{1}{2010^2}\right) \left(1-\frac{1}{2011^2}\right)==\frac{x 2011}{2} \]
I believe you posted the above, however, I think you meant to post:
\[\left(1-\frac{1}{2^2}\right) \left(1-\frac{1}{3^2}\right)\text{...} \left(1-\frac{1}{2010^2}\right) \left(1-\frac{1}{2011^2}\right)==\frac{x 2011}{2} \]
The above can be rewritten as:
\[\prod _{n=2}^{2011} \left(1-\frac{1}{n^2}\right)=\frac{x 2011}{2} \]
The product is equal to:
\[\prod _{n=2}^{2011} \left(1-\frac{1}{n^2}\right)=\frac{1006}{2011} \]
So the equation becomes:
\[\frac{1006}{2011}=\frac{x 2011}{2} \]
Solving for x,
\[x=\frac{2012}{4044121} \]