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I am still confued about the definition of an asymptotic series...

Let says a function f(x) is expanded as a series, Sn, and the remainder is Rn.

f(x) = Sn +Rn

so if we divide both sides by Sn, then

f(x)/Sn = 1+ Rn/Sn

So f(x)/Sn tends to 1 when Rn/Sn tends to zero as say x tends to c.

Why not an asymptotic series defined in this way?

I just can't accept that it is the remainder divided by the "last retained term in the series" say (x-c)^N in Sn, rather than the whole series Sn...

Could anyone help make this clear to me?

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# Aymptotic Series

Can you offer guidance or do you also need help?

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