Đặt \(S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+....+\frac{2013}{1+2013^2+2013^4}\)
Xét:
\(\frac{k}{k+k^2+k^4}=\frac{1}{2}\cdot\frac{k^2+k+1-k^2+k-1}{k^4+k^2+1}\)
\(=\frac{1}{2}\cdot\frac{k\left(k+1\right)+1-k\left(k-1\right)-1}{\left(k^2+1\right)^2-k^2}\)
\(=\frac{1}{2}\left[\frac{1}{k\left(k-1\right)+1}-\frac{1}{k\left(k+1\right)+1}\right]\)
Áp dụng :
\(S=\frac{1}{2}\left[\frac{1}{1\cdot0+1}-\frac{1}{1\cdot2+1}+\frac{1}{2\cdot1+1}-\frac{1}{2\cdot3+1}+.....+\frac{1}{2013\cdot2012+1}-\frac{1}{2013\cdot2014+1}\right]\)
\(=\frac{2027091}{4054183}\)
