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 http://www.voofie.com/concept/Mathematics/" [Broken]:

http://www.voofie.com/content/152/how-to-prove-eat-e-at_0-eat-t_0/" [Broken]

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.

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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:

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

fix question my question is
How prove this,
$$\exp(At)\exp(-At_0)=\exp(A(t-t_0))$$
using as above properties

http://www.voofie.com/content/152/how-to-prove-eat-e-at_0-eat-t_0/ [Broken]

but i dont understand how change sumatoria infinite to finite, Where i can read this?

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