I The sum of positive integers up to infinity: Was Sirinivasa right?

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The discussion centers on the sum of positive integers, concluding that the limit of the series diverges to infinity. It highlights the implications of allowing negative summands, where rearrangements can lead to different results, particularly in alternating series. An example is provided, showing that the series can converge to different values based on its arrangement. The conversation also touches on the importance of proper mathematical notation and the need for clarity in discussions about convergence. Ultimately, the thread emphasizes the complexity of series and the significance of adhering to mathematical rigor.
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What did you want to show? You could as well consider ##S_n=1+2+\ldots+n## and observe that ##S_{n+1}\geq S_{n}+1## for every ##n##, hence
$$
S:=\displaystyle{\lim_{n \to \infty}S_n}\geq \lim_{n \to \infty}(S_1 + n)=1+\lim_{n \to \infty}n = \infty .
$$

Things get interesting if you allow negative summands. In that case, re-orderings could result in different sums.
 
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Likes jedishrfu and Walid-yahya
Here is an interesting note on series with alternating signs:
$$
\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}\pm\ldots=\log 2
$$
which can be rearranged such that
$$
\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}\pm\ldots=\log \sqrt{2}
$$
(My notation here is sloppy since it doesn't show the rearrangement. It is only to emphasize that rearrangements aren't automatically allowed. The reference is precise at this point.)

Reference: https://www.physicsforums.com/insig...rom-zeno-to-quantum-theory/#Domains-of-Series
 
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Likes phinds, Walid-yahya and jedishrfu
Well, you have a sequence that is clearly not Cauchy; that itself should do it as proof that it doesn't converge.
 
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Great note dear It is clear that you have realized that the sum will reach infinity, and I place in your hands these papers for a new formula for this series.
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You can play with rearrangements of the natural numbers as often as you like, but this is nothing we could discuss here. Furthermore, please use ##\LaTeX## (https://www.physicsforums.com/help/latexhelp/) instead of uploading pictures.

Your last sentence is nonsense and suggests an assessment as a personal speculation which we do not discuss here. It is the third shortest way to leave our community. The theory of cardinalities is not trivial and the term "wave" in your post is nonsense, particularly on a website dedicated to physics.

This thread is closed now.
 
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