Question

Find the sum of the series `1+2x+3x^(2)+(n-1)x^(n-2))` using differentiation.

Answer 1

We know that `1+x+x^(2)+…+x^(n-1)=(1-x^(n))/(1-x).`
Differentiating both sides w.r.t.x, we get
`0+1+2x+3x^(2)+…+(n-1)x^(n-2)`
`=((1-x)(d)/(dx)(1-x^(n))-(1-x^(n))(d)/(dx)(1-x))/((1-x)^(2))`
`"or "1+2x+3x^(2)+...+(n-1)^(x-2)=(-(1-x)nx^(n-1)+(1-x^(n)))/((1-x)^(2))`
`"or "1+2x+3x^(2)+...+(n-1)^(x-2)=(-nx^(n-1)+(n-1)x^(n)+1)/((1-x)^(2))`