Question

Use differentiation to find a power series representation of f(x) =1/(8+x)^2. State its radius of convergence.

Answer 1

Consider the antiderivative of \(f\) first:
\[f(x)=\frac{1}{(8+x)^2}~~\implies~~F(x)=\int f(x)~dx=-\frac{1}{8+x}+C\]
Rewrite this expression (I'll ignore the \(C\) from here; it'll disappear when we take the derivative again):
\[F(x)=-\frac{1}{8\left(1-\left(-\dfrac{x}{8}\right)\right)}\]
Does this expression look like it has a familiar type of power series? (The answer should be yes, but I suppose that's up to you.)