$\begin{array}{l}\phantom{\phantom{;}n\phantom{;}+1;}{\phantom{;}n^{7}-n^{6}+n^{5}-n^{4}+n^{3}-n^{2}+n\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}n\phantom{;}+1\overline{\smash{)}\phantom{;}n^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}n\phantom{;}+1;}\underline{-n^{8}-n^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-n^{8}-n^{7};}-n^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n;}\underline{\phantom{;}n^{7}+n^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}n^{7}+n^{6}-;x^n;}\phantom{;}n^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n-;x^n;}\underline{-n^{6}-n^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-n^{6}-n^{5}-;x^n-;x^n;}-n^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}n^{5}+n^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}n^{5}+n^{4}-;x^n-;x^n-;x^n;}\phantom{;}n^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-n^{4}-n^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-n^{4}-n^{3}-;x^n-;x^n-;x^n-;x^n;}-n^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}n^{3}+n^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;\phantom{;}n^{3}+n^{2}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}n^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-n^{2}-n\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;-n^{2}-n\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}-n\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}n\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}n\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;;;;;;;\phantom{;}n\phantom{;}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2\phantom{;}\phantom{;}\\\end{array}$

$n^{7}-n^{6}+n^{5}-n^{4}+n^{3}-n^{2}+n-1+\frac{2}{1+n}$