28.31. Compute the sum $1+2 x+3 x^{2}+\ldots+n x^{n-1}$ by differentiating the equality $1+x+x^{2}+\ldots+x^{n}=\frac{1-x^{n+1}}{1-x}$.

See also problems $14.16,30.14$.

## 28.6. Tangent and Normal
28.31. Answer: $\left(\frac{1-x^{n+1}}{1-x}\right)^{\prime}=\frac{1-(n+1) x^{n}+n x^{n+1}}{(1-x)^{2}}$.

Remark. Another calculation of the required sum is given in the solution to problem 9.6.