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Due several hints in the comments I modify my query. It turns out that I have some know-how to use calculus but almost no know-why regarding analysis. That is why I alter my question to is my assumption reasonable.

Why I ask? Recently I found two hints, term-wise integration would require some theoretical justification, first one in a comment here, another in an elderly inaugural lecture held in 1911 by Oskar Perron -- "Über Wahrheit und Irrtum in der Mathematik" (German: "About verity and fallacy within mathematics") -- containing the unambiguous statement (or warning?), that term-wise differentiation and integration will lead to wrong results if specific prerequisites lack.

For sure without any justification of any "specific prerequisites" I evolved as student the normal distribution as Maclaurin series, integrated it term-wise, and combined it (after a polinomial division) to following formula

$$Q(x)=\frac{1}{2}+x\frac{e^{-x^2/2}}{\sqrt{2\pi}}\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!!}$$

(which nowadays you find also in Wikipedia)

Today, after reading chapter 6 of Abbott's "Understanding Analysis" I assume following conditions should be checked to ensure "usability" of the approach:

  • is Normal Distribution convertable to a Maclaurin series? Yes, it is, since it's not "extremely flat" at $X=0$
  • how big is the "radius of convergence" (or interval of convergence)? -- I did not find out yet
  • when computed to an accuracy of 10..12 digits the remnant up to infinity (according Lagrange's Reminder Theorem) must be small enough to have no impact on the result

Question: is the verification of this three points sufficient? What else should be checked?

Note: commentator Sgg8 suggested Abel's theorem as only prerequisite. But I can't see (yet) the reasons.

Trivia: Some time ago I found by reverse engineering the firmware of an ancient HP21S pocket calculator that a. m. formula is used for input $|x|\le2.32$ while other models use approximations. (When HP used it too, maybe I was not all that wrong -- but who knows what they once checked before relying on it.) I assume the range $|x|\le 2.32$ is due to shortest execution time, thus the second algotithm is faster for $x\gt 2.32$

m-stgt
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  • Offhand I'd guess the comments refer to checking "uniform summability" or "uniform convergence of the series". This turns out to be automatic in the interval of convergence of a power series. (If it matters, this condition is sufficient but not necessary.) – Andrew D. Hwang Apr 27 '24 at 15:06
  • @AndrewD.Hwang -- TY, for "uniform convergence" I found a hint in Wikipedia, that "The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus", but for "uniform summability" I only found papers behind paywall as of yet. – m-stgt Apr 27 '24 at 15:30
  • @m-stgt it seems like so far you've only been dealing with Calculus, not Analysis. Calculus is rather about calculating stuff without understanding what's going on. Analysis is about rigorous defining every term (such as "integral", "infinite", "infinite sum") and proving every single fact you use (such as interchanging integral and infinite sum). For introductory Analysis, take a look at Spivak's "Calculus" or Abbott's "Understanding Analysis". The latter is more "user friendly", that is, gives more intuition. Spivak still gives you the intuition, covers way more material, has great problems – Sgg8 Apr 27 '24 at 19:16
  • @m-stgtIn the answer I linked, I prove that basically if the function can be represented as Maclaurin series, then you can always interchange sum and integral – Sgg8 Apr 27 '24 at 19:18
  • @m-stgt but there is NO WAY you'll learn everything necessary about the Riemann integration and series in 24 hours. Take your time, work on the books I mentioned. Getting comfortable with Analysis will usually take from 6 months to several years depending on how much time daily you can dedicate to that and on your prior mathematical experience – Sgg8 Apr 27 '24 at 19:21

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