<s>[INST] Use differentiation to find a power series Use differentiation to find a power series of
$f(x) = \frac{1}{(8+x)^2}$
$ f'(x) = \frac{-2}{(8+x)^3} $
how do I find the power series of this? I can not go next step.
 [/INST] $$
g(x) = \frac{1}{(8+x)} = \sum_{n=0}^{\infty} \frac{(-1)^nx^n}{8^{n+1}}
$$
Taking the derivative in both sides: (n=0 is constant)
$$
g'(x) = \frac{-1}{(8+x)^2} =  \sum_{n=1}^{\infty} \frac{n(-1)^nx^{n-1}}{8^{n+1}}
$$
$$
-f(x) = \frac{-1}{(8+x)^2} =  \sum_{n=0}^{\infty} \frac{(n+1)(-1)^{n+1}x^{n}}{8^{n+2}}
$$
$$
f(x) = \sum_{n=0}^{\infty} \frac{(n+1)(-1)^{n}x^{n}}{8^{n+2}}
$$
</s>