Review One

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