\(\frac{1}{2011x}+\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{2010.2011}=1\)
\(\Rightarrow\frac{1}{2011x}+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2010.2011}=1\)
\(\Rightarrow\frac{1}{2011x}+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2010}-\frac{1}{2011}=1\)
\(\Rightarrow\frac{1}{2011x}+1-1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2010}-\frac{1}{2011}=0\)
\(\Rightarrow\frac{1}{2011x}-\frac{1}{2011}=0\)
\(\Rightarrow2011x-2011=0\Rightarrow x=1\)
\(\frac{1}{2011x}+\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+....+\frac{1}{2010x2011}=1\)
=> \(\frac{1}{2011x}+\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+........+\frac{1}{2010x2011}\)= 1
=> \(\frac{1}{2011x}+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}+...........+\frac{1}{2010}-\frac{1}{2011}\)= 1
=> \(\frac{1}{2011x}+1-1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-...............+\frac{1}{2010}-\frac{1}{2011}\)= 0
=> \(\frac{1}{2011x}-\frac{1}{2011}=0\)
=> 2011x - 2011 = 0
=> x = ( 0 + 2011 ) : 2011 = 2011 : 2011 = 1