Pierādīt, ka

\[\frac{1}{1^{4}+1^{2}+1}+\frac{2}{2^{4}+2^{2}+1}+\frac{3}{3^{4}+3^{2}+1}+\ldots+\frac{2007}{2007^{4}+2007^{2}+1}<\frac{1}{2}\]