Calculating the Matrix Exponential

For a Bounded Linear Operator $X$, we are able to exploit the Taylor series expansion of the exponential $e^x={\sum }_{k=0}^{\infty }\frac{1}{k!}{x}^k$, to obtain the resulting formula $e^X={\sum }_{k=0}^{\infty }\frac{1}{k!}{X}^k$.