I have a function and I have to find the taylor series expansion using term by term integration and differentiation:
$$f(x) = \frac{1}{x^2} \quad x_0 =3$$
Well, my concern is that I only know the general formula
$$\sum \frac{f^{(n)}(x_0)}{n!}\cdot(x-x_0)^n$$
Which is the easiest, but not allowed to use, but how should I proceed using term-by-term approach? I failed to find some real examples, only general theory (that's why I took the simplest example to understand the idea).
Notice that $$f(x)=\frac{d}{dx}\left(\frac{1}{-x}\right)=\frac{d}{dx}\left(\frac{1}{-3-(x-3)}\right)=-\frac{1}{3}\frac{d}{dx}\left(\frac{1}{1+(x-3)/3}\right).$$ But $$ \frac{1}{1+(x-3)/3}=\sum_{k=0}^\infty\left(\frac{-1}{3}\right)^k(x-3)^k. $$ Hence, for each $x\in\mathbb{R}$ such that $|x-3|<3$, $$ f(x)=-\frac{1}{3}\sum_{k=1}^\infty\left(\frac{-1}{3}\right)^kk(x-3)^{k-1}=\sum_{k=1}^\infty\left(\frac{-1}{3}\right)^{k+1}k(x-3)^{k-1}. $$