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What is the sum of all real numbers?

Note by Adharsh M
1 month ago

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which wiki, which section, please explain Adharsh M · 3 weeks, 5 days ago

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@Adharsh M The absolutely convergent wiki that I linked to. Read through the entire wiki, esp since it sounds like you are unfamiliar with this concept. Calvin Lin Staff · 3 weeks, 5 days ago

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why doesn't it work? Adharsh M · 3 weeks, 5 days ago

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@Adharsh M Read the wiki, esp the last section. Calvin Lin Staff · 3 weeks, 5 days ago

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Mr.Calvin Lin, could you please explain, what actually you want to convey. Adharsh M · 3 weeks, 5 days ago

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@Adharsh M The sum of all real numbers is undefined. This is because the summation is not absolutely convergent, and hence we cannot assign a value to it.

Why does the naive approach of "pairing up \(x\) with \(-x\) to make the sum of infinitely many \( x -x = 0\)" not work? Calvin Lin Staff · 3 weeks, 5 days ago

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if I say that all the real numbers will have their negative pair ,example 5 and -5,so similarly for infinity there will be -infinity which will sum to 0 Adharsh M · 3 weeks, 6 days ago

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@Adharsh M That's a good thought. Under what scenarios can we do? (Maybe the question to answer first is: what are we doing here?) Calvin Lin Staff · 3 weeks, 6 days ago

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is it 0? Adharsh M · 1 month ago

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@Adharsh M It's \(\infty\). Munem Sahariar · 1 month ago

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@Munem Sahariar Can you justify why? Calvin Lin Staff · 1 month ago

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@Calvin Lin Because there are infinitely many real numbers. Munem Sahariar · 1 month ago

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@Munem Sahariar And so? Calvin Lin Staff · 1 month ago

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@Calvin Lin Since there are infinitely many real numbers or uncountably many real numbers so we can't determine there sum. Munem Sahariar · 1 month ago

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@Munem Sahariar So, you are changing your answer from "\( \infty\)" to "undefined"? Calvin Lin Staff · 1 month ago

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@Calvin Lin No. \(\infty\) and undefined both are my answers. Munem Sahariar · 4 weeks, 1 day ago

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@Munem Sahariar It is very rare for an equation to have 2 answers.

At the very most, it would be "Under different interpretations". For example, \( 1 + 2 + 3 + \ldots \) would be infinity in the usual calculus treatment of infinite sums, but could be \( - \frac{1}{12} \) under the analytic continuation of \( \sum n^s \). As such, can you clarify under which interpretation we get the answer of "\(\infty\)", and under which interpretation we get the answer of "undefined"? Calvin Lin Staff · 3 weeks, 6 days ago

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@Adharsh M No, it is not. Calvin Lin Staff · 1 month ago

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