I apologize if this is a rather lame question, but I've always been a little touchy with my partial fraction decompositions and I'm hoping to get better at them. Could you verify (or correct?) my decomposition below?
I've been asked to find $\mathcal{L}^{-1}\left\{\frac{6}{\left(s-1\right)^4}\right\},$ and I'm certain that one way to find it is break it into partials. My textbook says such a decomposition looks like \begin{align} \frac{1}{\left(s-r\right)^m}=\frac{A_1}{s-r}+\frac{A_2}{\left(s-r\right)^2}+\dots+\frac{A_m}{\left(s-r\right)^m},\tag{1} \end{align} but then they don't provide any "simple" examples with just $\left(s-r\right)^m$ in the denominator, they always mix it with things. Would I be correct in saying then, that the above fraction can be written as \begin{align} \frac{6}{\left(s-1\right)^4}=\frac{A_1}{\left(s-1\right)}+\frac{A_2}{\left(s-1\right)^2}+\frac{A_3}{\left(s-1\right)^3}+\frac{A_4}{\left(s-1\right)^4}?\tag{2} \end{align}