Interpreting .
e^{At} = \sum\_{0}^{\infty}\frac{(At)^n }{n!} = I + At + \frac{1}{2}(At)^2 +\dots$$ Referencing [[Eigenmatrices]] and used in linear-algebra [[Differential Equations with Constant Coefficients|differential equations]], this visualises the act of raising a number to a matrix power.e^{\Lambda t} = \begin{bmatrix} e^{\lambda_{1} t} & & \ & \ddots & \ & & e^{\lambda_{n} t} \end{bmatrix}
### Geometric Series\frac{1}{1-x} = \sum_{0}^{\infty} x
This convenient series can be used to approximate any inverse $(I-At)^{-1}$.(I-At)^{-1} \approx I + At + (At)^2 + \dots
For small values of $t$ the bigger terms of the sum shrink quickly and only the first few ($n = 1, 2, 3$) approximate the result well.