This was something that popped up in class and confused me...
So knowing that the power series may be differentiated term by term inside the interval of convergence, using the Maclaurin series you can derive the differentiation formula for the function $f(x) = \frac{1}{1-x}$.
What do they mean by differentiation formula and how do I get it using Maclaurin series? I'm quite confused xD
The McLaurin formula is just the Taylor expansion calculated in $0$, so: $$ f(x)=f(0)+f'(0)x+f''(0)\frac{x^2}{2}+\dots=\sum_{k=0}^\infty f^{(k)}(0)\frac{x^k}{k!}. $$
In your case: $$ f(0)=1, $$ $$ f'(x)=-\frac{1}{(1-x)^2}\cdot(-1); f'(0)=1, $$ and so on.
It says differentiation formula, as you have to compute all the derivatives of $f(x)$.