$\begin{array}{l}\phantom{\phantom{;}2x\phantom{;}-1;}{\phantom{;}512x^{9}+256x^{8}+128x^{7}+64x^{6}+32x^{5}+16x^{4}+8x^{3}+4x^{2}+2x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;}2x\phantom{;}-1\overline{\smash{)}\phantom{;}1024x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}2x\phantom{;}-1;}\underline{-1024x^{10}+512x^{9}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-1024x^{10}+512x^{9};}\phantom{;}512x^{9}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n;}\underline{-512x^{9}+256x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-512x^{9}+256x^{8}-;x^n;}\phantom{;}256x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n;}\underline{-256x^{8}+128x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-256x^{8}+128x^{7}-;x^n-;x^n;}\phantom{;}128x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n;}\underline{-128x^{7}+64x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-128x^{7}+64x^{6}-;x^n-;x^n-;x^n;}\phantom{;}64x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n;}\underline{-64x^{6}+32x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-64x^{6}+32x^{5}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}32x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-32x^{5}+16x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;-32x^{5}+16x^{4}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}16x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-16x^{4}+8x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;-16x^{4}+8x^{3}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}8x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-8x^{3}+4x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;;;-8x^{3}+4x^{2}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}4x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-4x^{2}+2x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;;;-4x^{2}+2x\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2x\phantom{;}\phantom{-;x^n}\\\phantom{\phantom{;}2x\phantom{;}-1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-2x\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;;;;;;;;;-2x\phantom{;}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}1\phantom{;}\phantom{;}\\\end{array}$

$512x^{9}+256x^{8}+128x^{7}+64x^{6}+32x^{5}+16x^{4}+8x^{3}+4x^{2}+2x+1+\frac{1}{2x-1}$