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**arze** Find the sum S(x) of the series $\displaystyle 1+2x+3x^2+...+(n+1)x^n$ by finding $\displaystyle (1-x)S(x)$.

I multiplies the series by (1-x) and got $\displaystyle 1+x+x^2+...+x^{n-1}$ Mr F says: This is wrong. It's $\displaystyle {\color{red}1+x+x^2+...+x^n - (n+1) x^{n+1}}$.

And the sum of this would be $\displaystyle \frac{1(1-x^n)}{(1-x)}$

Then what would S(x) be? $\displaystyle \frac{1(1-x^n)}{(1-x)}.\frac{1}{(1-x)}$?

The answer is supposed to be $\displaystyle \frac{[1-(n+2)x^{n+1}+(n+1)x^{n+2}]}{(1-x)^2}$

Thanks