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

$\dfrac1{1-x}=\displaystyle\sum_{n\ge0}x^n$

which has derivative

$\dfrac1{(1-x)^2}=\displaystyle\sum_{n\ge0}nx^{n-1}=\sum_{n\ge0}(n+1)x^n$

Then

$f(x)=\dfrac1{(10+x)^2}=\dfrac1{100}\dfrac1{\left(1-\left(-\frac x{10}\right)\right)^2}=\frac1{100}\displaystyle\sum_{n\ge0}(n+1)\left(-\frac x{10}\right)^n$

which converges

$\left|-\dfrac x{10}\right|<1\implies|x|<10$

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