+9479 \(\phantom{=\quad}\dfrac{x^2+2x^4+3x^6+\cdots+1005x^{2010}}{2x+4x^3+6x^5+\cdots+2010x^{2009}}\)
Factor x out of all the terms in the numerator and factor 2 out of all the terms in the denominator to get:
\({=\quad}\dfrac{x(x+2x^3+3x^5+\cdots+1005x^{2009})}{2(x+2x^3+3x^5+\cdots+1005x^{2009})}\)
Now we can cancel out the common factor from the numerator and denominator to get:
\({=\quad}\dfrac{x}{2}\)
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