Dispelling limit confusions and cheating

3,547 views Jun 19, 2024

There are serious confusions about the role of "limits" in pure mathematics, and in this video we try to clarify the difficulties that are wide-spread in the subject, but purposely unacknowledged. We do this by explaining why there are really three, not two, kinds of "limiting" behaviour of "infinite series" in analysis. The notion of a series satisfying the Cauchy condition plays an important role. Video Contents: Three kinds of limits for series Converting a series to a sequence Limit of a series/sequenc The Cauchy conditio False fact re convergence of Cauchy sequences The big cheat: creating limits out of thin air *********************** My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses are being developed at openlearning.com. The first one, already underway, is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon, / njwildberger Your support would be much appreciated. Here are the Insights into Mathematics Playlists: • MathHistory: A course in the History ... • WildTrig: Intro to Rational Trigonometry • Math Foundations • Wild Linear Algebra • Famous Math Problems • Box Arithmetic • Elementary Mathematics (K-6) Explained • Year9Maths • Ancient Mathematics • Wild West Banking • Sociology and Pure Mathematics • Sociology and Pure Physics • Old Babylonian Mathematics • Probability and Statistics: an introd... • Boole's Logic and Circuit Analysis • Universal Hyperbolic Geometry • Differential Geometry • Algebraic Topology • MathSeminars • Playing Go • Diffusion Symmetry: A bridge between ...Here are the Wild Egg Maths Playlists (some available only to Members!) • Algebraic Calculus One • Classical to Quantum (Members only)[ • Algebraic Calculus Two](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-) • Advice for mathematics exploration (M... • Solving Polynomial Equations (Members...[ • The Hexagrammum Mysticum (Members only)](https://www.youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_) • Algebraic Calculus and dCB curves (Me... • Dynamics on Graphs (Members only) • Playlist • Triangle Geometry • Explorations with q-series (Members o... • Six: An elementary course in Pure Mat... • Maxel Inverses and Orthogonal Polynom...************************

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[Music] Hello everyone, we're talking about the sociology of pure mathematics. In my

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last few videos, I've received many comments symptomatic of a deep confusion spread across

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the subject, particularly around limits, real numbers, sequences, and series. Today,

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I'll make an elementary video on this topic to clarify some of these confusions and actually

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identify some "cheating" that's going on. Cheating is a strong word, but I believe it's largely

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unconscious. At some level, most professional mathematicians are aware of it, even if they

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struggle to acknowledge it fully. This issue has an interesting sociological aspect, where a

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logical difficulty gets embedded in the subject, and everyone sort of ignores it. While I can't

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shed too much light on the psychology or sociology of this phenomenon, I will present

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the difficulties, which are quite straightforward once understood.

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In classical analysis, there are typically two kinds of series behavior: convergence and

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divergence. However, there's actually a third kind, represented by these three examples:

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The first series is simply the sum of natural numbers:

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1 +

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3 +

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5 +

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7 +

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… 1+3+5+7+…. The sequence of

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partial sums forms squares: 1, 4, 9, 16, and so on. This series clearly diverges because

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the sums grow without bound. We often denote this by saying the series tends to infinity,

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but there's no actual number called "infinity."

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The second series, 1

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2 +

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1 4

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8 +

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… 1+

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2 1

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4 1

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8 1

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+…, is a familiar geometric

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series with a common ratio of 1

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2 2

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. The partial sums approach 2, leading us to say the series converges to 2.

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The third series involves the reciprocals of factorial cubes:

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1 +

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1 1

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! 3

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2 !

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3 +

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1 3

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! 3

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…

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1+ 1!

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3 1

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2! 3

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3 1

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+…. It appears to converge to a number around

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2.1297, but this convergence is more problematic and not as straightforward as in the second case.

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This series illustrates a different kind of convergence, which is crucial to understand.

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In classical analysis, we transform a series into a sequence of partial sums to determine

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if it converges. If these partial sums approach a limiting value, we say the series converges.

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This process involves checking if the terms 𝑎

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𝑘 a

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of the series (the differences between consecutive partial sums) approach zero as

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𝑘 k increases.

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The Cauchy condition adds a more sophisticated layer, stating that for any given small number

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1 𝑚

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m 1

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, there exists a point

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𝑘 k beyond which all subsequent

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partial sums are within 1

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𝑚 m

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of each other. This condition ensures not just that the terms

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𝑎 𝑘

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a k

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go to zero,

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but that the partial sums become uniformly close to each other.

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However, a significant issue arises: the converse of this condition is often incorrectly taught.

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It's not always true that if a sequence of partial sums satisfies the Cauchy condition,

Converting a series to a sequence

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then it converges to a real number. This misconception leads to a form of

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"cheating," where we assume the existence of a real number limit without truly establishing it.

Limit of a series/sequence

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The third example, the series of reciprocal factorial cubes, exemplifies this issue.

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While the partial sums appear to converge to a certain value, there's no guarantee

False fact re convergence of Cauchy sequences

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that this value is a real number. The supposed limit of this series might not be accessible

The big cheat: creating limits out of thin air

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in any concrete way other than through the ongoing computation of the series itself.

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This problem illustrates a critical point about real numbers in mathematics: many real

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numbers are defined only as the end result of an infinite process, which can be seen as a kind of

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logical cheat. This idea is often not clearly communicated to students, leading to confusion.

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In summary, there are three types of series behaviors: divergence, simple convergence,

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and a more complex, problematic convergence. Understanding the nuances of these behaviors is

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crucial for a deeper understanding of mathematical analysis and the nature of real numbers.

Dispelling limit confusions and cheating

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