- #1
Walid-yahya
- 2
- 0
The sum of positive integers up to infinity: Was Sirinivasa right?
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Discussion Overview
The discussion revolves around the concept of summing positive integers to infinity, exploring various mathematical perspectives and implications, including convergence, rearrangements of series, and the implications of negative summands. The scope includes theoretical considerations and mathematical reasoning.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant suggests that the sum of positive integers diverges to infinity, using the notation ##S_n=1+2+\ldots+n## and observing that ##S_{n+1} \geq S_n + 1##.
- Another participant introduces the concept of alternating series, noting that rearrangements can lead to different sums, specifically referencing the series that converges to ##\log 2## and can be rearranged to converge to ##\log \sqrt{2}##.
- A different participant argues that a sequence not being Cauchy serves as proof that it does not converge.
- One participant acknowledges the divergence to infinity and offers papers on a new formula related to the series.
- Another participant expresses skepticism about the relevance of rearrangements of natural numbers and critiques a previous comment as personal speculation, emphasizing the complexity of cardinalities and dismissing the term "wave" as nonsensical.
Areas of Agreement / Disagreement
Participants express differing views on the convergence of the sum of positive integers and the implications of rearranging series. There is no consensus on the validity of certain claims, and the discussion remains unresolved.
Contextual Notes
Some arguments depend on the definitions of convergence and the treatment of series with rearrangements. The discussion also touches on the implications of negative summands, which introduces additional complexity.
- #2
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- 28,452
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.
$$
S:=\displaystyle{\lim_{n \to \infty}S_n}\geq \lim_{n \to \infty}(S_1 + n)=1+\lim_{n \to \infty}n = \infty .
$$
Things get interesting if you allow negative summands. In that case, re-orderings could result in different sums.
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- #3
- 20,815
- 28,452
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
$$
\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n}=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}\pm\ldots=\log 2
$$
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
- #4
WWGD
Science Advisor
Homework Helper
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Well, you have a sequence that is clearly not Cauchy; that itself should do it as proof that it doesn't converge.
- #5
Walid-yahya
- 2
- 0
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.
- #6
- 20,815
- 28,452
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.
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.
Last edited:
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