What is the name of the following property of summation (t is the time)?
$$\sum_{i=a}^{b}x_{i}(t)=\sum_{i=-b}^{-a}x_{-i}(t)$$
Could you show me a proof of it please?
Thank you.
PS: if possible show me webpages of the above property please.
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This only works if $x(i)$ is an even indexing function. Cf. the same property for integrals. There really isn't a name. – Sean Roberson Jun 14 '17 at 17:32
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@SeanRoberson No, you missed the change of sign internally. – Thomas Andrews Jun 14 '17 at 17:33
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My mistake, I see it now. – Sean Roberson Jun 14 '17 at 17:33
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I'd prove it by induction on $b-a$, since $\sum_{i=a}^{b}$ is essentially defined inductively on $b-a$. – Thomas Andrews Jun 14 '17 at 17:36
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I doubt this specific equality has a name, but it is a case of the general commutativity law of addition. – Thomas Andrews Jun 14 '17 at 17:37
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I don't think property has a special name. It's just what happens when you change indices. Specifically, changing the index $i$ to $-i$ in this case. – Jun 14 '17 at 17:39
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This can be called the "rearrangement property of addition". I.e., it doesn't matter what order you add together the same finite set of numbers, you will get the same result. It follows by easy induction from the commutative and associative properties of addition.
What you have to recognize is that under all the notation, what you have here is the same sum expressed in two different ways, except reversed in order.
The LH sum when written out is (dropping the superfluous $(t)$): $$x_a + x_{a+1} + x_{a+2} + \dots + x_{b-2} + x_{b-1} + x_b$$
The RH sum is $$x_{-(-b)} + x_{-(-b+1)} + x_{-(-b+2)} + \dots + x_{-(-a-2)} + x_{-(-a-1)} + x_{-(-a)}\\=x_b + x_{b-1} + x_{b-2} + \dots + x_{a+2} + x_{a+1} + x_a$$
Paul Sinclair
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