Question

Find a power series representation for f(x) = 1 (10 + x)2 . f(x) = ∞ n = 0 What is the radius of convergence, R? R = (b) Use part (a) to find a power series representation for f(x) = x3 (10 + x)2 . f(x) = ∞ n = 0 What is the radius of convergence, R?

Answer 1

a. Recall that for
`|x|<1`, we have

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

which has derivative

`\frac1{(1-x)^2}=\displaystyle\sum_(n\ge0)nx^(n-1)=\sum_(n\ge0)(n+1)x^n`

Then

`f(x)=\frac1{(10+x)^2}=\frac1{100}\frac1{\left(1-\left(-\frac x{10}\right)\right)^2}=\frac1{100}\displaystyle\sum_(n\ge0)(n+1)\left(-\frac x{10}\right)^n`

which converges

`\left|-\frac x{10}\right|<1\implies|x|<10`

b. From the above result, it's evident that the radius of convergence is
`R=10`.