Với x = 1 thì [TEX]S=2n[/TEX]
Với x khác 1 ta có:
[tex]S=(x+x^2+...+x^n)+(\frac{1}{x}+\frac{1}{x^2}+...+\frac{1}{x^n})=\frac{x^{n+1}-1}{x-1}+\frac{\frac{1}{x^{n+1}}-1}{\frac{1}{x}-1}=\frac{x^{n+1}-1}{x-1}+\frac{\frac{1-x^{n+1}}{x^{n+1}}}{\frac{1-x}{x}}=\frac{x^{n+1}-1}{x-1}+\frac{x^{n+1}-1}{(x-1)x^n}=\frac{(x^n+1)(x^{n+1}-1)}{x^n(x-1)}[/tex]
Với x = 1 thì [TEX]S=2n[/TEX]
Với x khác 1 ta có:
[tex]S=(x+x^2+...+x^n)+(\frac{1}{x}+\frac{1}{x^2}+...+\frac{1}{x^n})=\frac{x^{n+1}-1}{x-1}+\frac{\frac{1}{x^{n+1}}-1}{\frac{1}{x}-1}=\frac{x^{n+1}-1}{x-1}+\frac{\frac{1-x^{n+1}}{x^{n+1}}}{\frac{1-x}{x}}=\frac{x^{n+1}-1}{x-1}+\frac{x^{n+1}-1}{(x-1)x^n}=\frac{(x^n+1)(x^{n+1}-1)}{x^n(x-1)}[/tex]