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fermatslibrary's profile
Fermat's Library
Fermat's Library
Fermat's Library
@fermatslibrary

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Fermat's Library

@fermatslibrary

A platform for illuminating academic papers. We publish an annotated paper every week. Our chrome extension for arXiv: https://fermatslibrary.com/librarian 

fermatslibrary.com
Joined September 2015

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    Fermat's Library‏ @fermatslibrary 10 Sep 2018
    • Report Tweet
    • Report NetzDG Violation

    Here's an obvious theorem that is actually false: One can arbitrarily rearrange the terms in a convergent series without changing its value. Counterexample: ∑(-1)ⁱ/i

    5:55 AM - 10 Sep 2018
    • 146 Retweets
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    • haxton Vikram Dutt Dimitrios Kalemis Mohamad Adami Ák 🐢 Christopher Schultes Darshan  J DJ 𝔸𝗅𝖾𝗑𝖺𝗇𝖽𝗋𝖾 𝕃𝖾𝗆𝗈𝗋𝗍 Sam
    18 replies 146 retweets 697 likes
      1. Tamás Görbe‏ @TamasGorbe 10 Sep 2018
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        Replying to @fermatslibrary

        Conditionally convergent series are convergent, but not absolute convergent. Example: 1-1/2+1/3-1/4+...=ln(2), but 1+1/2+1/3+1/4+...=∞. Riemann proved that the terms of such a series can be rearranged so that the new series converges to an arbitrary number, or even diverges.

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      1. Robert Low‏ @RobJLow 10 Sep 2018
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        Replying to @fermatslibrary

        Yes: the commutativity of finite sums doesn't imply it for infinite ones. This is like each term in a convergent sequence having a property, but the limit not having it.

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      2. Yannayl‏ @Yannayli 10 Sep 2018
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        Replying to @BitterSiPeter @fermatslibrary

        Actually you can rearrange the series to converge to any real number (and infinities ) or diverge. Intuitively: fix M and start adding positive numbers until the partial sum is greater than M, then add negative numbers until the sum is smaller than M. Repeat.

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      1. Yannayl‏ @Yannayli 10 Sep 2018
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        Replying to @fermatslibrary

        https://en.m.wikipedia.org/wiki/Riemann_series_theorem …

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      2. aiun‏ @KivrinAiun 10 Sep 2018
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        Replying to @trn5cnd @fermatslibrary

        i'm not either, but i like to learn also i'm not sure why they went with a weird counterexample; if i understand correctly there are easier ones (https://math.stackexchange.com/questions/2151423/how-come-rearrangement-of-a-convergent-series-may-not-converge-to-the-same-value …)

        1 reply 1 retweet 0 likes
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      1. Chris Dalla Diva‏ @cdr43210 10 Sep 2018
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        Replying to @fermatslibrary

        @Matt_Meehan107 hmmm...

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      1. ville‏ @villeuo 10 Sep 2018
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        Replying to @fermatslibrary

        ie. conditional convergence

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      1. Per Lindholm‏ @perrabyte 10 Sep 2018
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        Replying to @fermatslibrary

        In machine learning you can still get the same recognition values. Like a rearranged image.

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