How does this nursery rhyme pertain to power series: “There was a little girl Who had a little curl Right in the middle of her forehead…"
How does this nursery rhyme pertain to power series: “There was a little girl Who had a little curl Right in the middle of her forehead…"
This is from The Way of Analysis by Strichartz, chapter $7$, section $7.4$, page $276$. He writes
"In discussing power series it is good to recall a nursery rhyme:"
"There was a little girl
Who had a little curl
Right in the middle of her forehead
When she was good
She was very, very good
But when she was bad
She was horrid."
I don't understand why or how this nursery rhyme is related to power series.
Kudos to the comment by Barry Cipra. The rest of the poem can be seen here. I wonder if the rest of the poem can be turned into having some mathematical significance.
Charlie Brown is a girl now?
– Henning Makholm
Aug 22 at 18:16
Perhaps the point is that the coefficients of a power series are either very, very easy to describe, or else very, very difficult.
– Blue
Aug 22 at 18:19
@MostafaAyaz This is the very beginning of the section on power series. There is no power series material before this.
– Al Jebr
Aug 22 at 18:23
Incidentally, the quoted lines are not a nursery rhyme (in the traditional "Mother Goose" sense). They are the first stanza of a children's poem by Henry Wadsworth Longfellow.
– Barry Cipra
Aug 22 at 19:02
1 Answer
1
The nursery rhyme may be referring to the fact that power series are "very, very good" within their radius of convergence (absolute convergence, term-by-term differentiation and integration, uniform convergence, etc.) while they diverge outside the radius of convergence (which is "horrid" as far as series are concerned).
The curl in the middle of her forehead may be referencing the boundary of convergence; on one side, it's very nice while on the other side, it's horrible. (It's also a possibility that it's simply part of a preexisting rhyme, as @Blue puts it.)
There's also the fact that a region of convergence of a power series has to be almost symmetric, so power series are no good in certain situations which require an asymmetric type of function. Whereas asymptotic power series don't have the same limitation. (e.g. Reportedly, there's a proof using this idea that a solution to the Standard Model, as an asymptotic power series in the fine structure constant, cannot be a convergent power series.)
– Daniel Schepler
Aug 22 at 18:51
+1. This captures the good-vs-bad notion better than my comment. As for the curl ... I don't think it's a direct reference to anything; it's just part of the pre-existing rhyme.
– Blue
Aug 22 at 18:53
In any case, authors of textbooks and other technical books sometimes do quote something non-technical before starting a new chapter, implying that the quote has some metaphorical relationship to the topics about to be discussed. This answer fits that pattern perfectly and I think is the most likely reason for including the poem in the book.
– David K
Aug 22 at 18:58
Outside the closure of its circle of convergence, the power series is not merely bad ( non-convergent), it is horrid: The sequence of terms of the series is unbounded.
– DanielWainfleet
Aug 23 at 13:31
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Can you add up some more detain such as an image of the chapter?
– Mostafa Ayaz
Aug 22 at 18:15