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For the definition of a finite series, I've seen two notations, i.e.

$~~~~~$Let $x_n=n$ for $n=1,...,N$.

$~~~~~$Let $x_n=n$ for $n\in\{1,...,N\}$.

Is there any difference in the meaning? Does it work also with "with"? I.e.

$~~~~~$Let $x_n=n$ with $n\in\{1,...,N\}$.

or is this incorrect?

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    $\begingroup$ If you use 'where' instead of 'with', it reads better. The first two are the same. $\endgroup$
    – vvg
    Sep 30, 2020 at 10:22

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I am not sure if 'with' can be used but the first two are the same, no difference. I don't recall reading the third one with 'with' ever, so it is definitely less popular, if not wrong.

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  • $\begingroup$ Would you write "Let $x_n$ with $n\in\mathbb{N}$ denote all ..." or would you stick with "Let $x_n$ for $n\in\mathbb{N}$ denote all ..."? Or, use "where"? $\endgroup$
    – Its_me
    Sep 30, 2020 at 12:41
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    $\begingroup$ I would use either 'for' or 'where'. $\endgroup$
    – ModCon
    Oct 1, 2020 at 10:31

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