Question

Express 15/(1 - 3x)2 as a power series by differentiating the equation below. what is the radius of convergence?

Answer 1

Recall that


`\displaystyle\frac1{1-3x}=\sum_(n\ge0)(3x)^n`

for
`|3x|<1`. Notice that


`(\mathrm d)/(\mathrm dx)\frac1{1-3x}=\frac3{(1-3x)^2}`

so that we have


`\displaystyle(15)/((1-3x)^2)=15(\mathrm d)/(\mathrm dx)\sum_(n\ge0)(3x)^n`

`\implies\displaystyle(15)/((1-3x)^2)=15\sum_(n\ge1)n(3x)^(n-1)`

(again, for
`|3x|<1`)