Property Function exp

How can prove this

$$\exp(At)\exp(-At_0)=\exp(A(t-t_0))$$?

using $$\displaystyle\sum_{i=0}^n{(1/k!)A^kt^k}$$

and this properties
in t=0
$$[\exp(At)]_{t} = I$$

$$exp(At)exp(-At)=I$$
$$\frac{dexp(At)}{dt}=Aexp(At)=exp(At)A$$
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 For your first equation, please refer to this question in Voofie/Mathematics: How to prove e^At e^(-At_0) = e^A(t-t_0)? I think you typed wrong in this formula: $$exp(At)_t=0 = I$$ 0 is not equal to I. And your what's your meaning of $$exp(At)_t$$? For this one $$exp(At)exp(-At)=I$$, you can use my result to prove easily. For the last one, you should try to use the power series expansion and differentiate term by term. You will get the answer easily too.

 Quote by juaninf How can prove this $$exp(At)_t=0 = I$$
To conclude, i suppose you mean

$$[\exp(At)]_{t=0} = I$$

Well, it's pretty simple:

$$[\exp(At)]_{t=0} = \left[\sum_{k=0}^\infty\frac{A^kt^k}{k!}\right]_{t=0}=I+0+0+\cdots=I$$

Property Function exp

fix question my question is
How prove this,
$$\exp(At)\exp(-At_0)=\exp(A(t-t_0))$$
using as above properties
 Thank,I am reading this web http://www.voofie.com/content/152/ho...-at_0-eat-t_0/ but i dont understand how change sumatoria infinite to finite, Where i can read this?