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Can anyone explain me why " a power series is asymptotic to a function" is defined as shown in the picture?

Referring to the first definitoin, it means that the remainder after N terms should be much smaller than (x-xo)^N? Actually, I can't "sense" the meaning of such definition.

Maybe, sometimes, a definition may not bare any meaning....

But, "asymptotic", in my mind, means "closer and closer" to something. And I guess this defintion should bare some implication for this. Can anyone explain this defintion for me?

I am completely stucked with this..Please kindly help.

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# Definition of Asymptoticity to a Function

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