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I'm reading a book introducing gravity and find something I don't understand. Please check the attached image. The sentence in red bracket claims that if one adding up gravitational fields of infinitely many objects, the total gravitational field depends on the order in which we add them.

I'm just wondering how do summation order affect the total gravitational field? Is this some mathematical magic in infinite series?enter image description here

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  • $\begingroup$ I assume this is a reference to the fact the Madelung Constant is conditionally convergent. $\endgroup$
    – jacob1729
    Commented Jul 19, 2019 at 20:24
  • $\begingroup$ The issue pointed out in the text has been discussed on this site here and here. $\endgroup$
    – knzhou
    Commented Jul 19, 2019 at 20:29
  • $\begingroup$ @jacob1729 Thank you. In Madelung constant case there exists both positive and negative charges so I can understand why order matters. But for gravity the masses should all be positive $\endgroup$
    – shinotakatoshi
    Commented Jul 19, 2019 at 20:34
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    $\begingroup$ It's because the gravitational forces due to these masses can be in different directions, and hence cancel. $\endgroup$
    – knzhou
    Commented Jul 19, 2019 at 20:38
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    $\begingroup$ Please do not post images of texts you want to quote, but type it out instead so it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$
    – ACuriousMind ♦
    Commented Jul 19, 2019 at 20:49

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