isn't the answer x divided by 2.

\(\frac{x^2+2x^4+3x^6+...+1005x^{2010}}{2x+4x^3+6x^5+...+2010x^{2009}}\)Simplify

Pretty nice question. Try to see the pattern!

\(\dfrac{\sum \limits_{k=1}^{1005}~k x^{2k}}{\sum \limits_{j=1}^{1005}~2jx^{2j-1}}=\\ \dfrac{x}{2}\dfrac{\sum \limits_{k=1}^{1005}~2k x^{2k}}{\sum \limits_{j=1}^{1005}~2jx^{2j}}=\\ \dfrac x 2\)