I am computing the exponential of a matrix via Taylor expansion to prove the end-result with induction.
For a matrix $A=PDP^{-1}$ where $D$ is the diagonalized matrix, is there any kind of formula for adding their sums like in the instance of
$$I+PDP^{-1}+\frac{PD^{2}P^{-1}}{2!}+\frac{PD^{3}P^{-1}}{3!}...?$$