You may have missed my message on another thread:
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If you are just now studying Power Series, you are by my count about 23 courses prior to where renormalization will be discussed. I think you're going to have to accept that the answers you get will be kind of hand-wavy.
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If you picked up a murder mystery, read a few pages, and then jumped to the end, you shouldn't be surprised if the killer's identity seems to make no sense.