Curiosity on this infinite product

In summary, the conversation discusses the concept of infinite products and the curious result of setting the infinite case to a variable, which leads to a contradiction. The conversation explores the possibility of a conceptual reason for this result and concludes that it is a fake math created by assuming a non-existent limit. The conversation also considers other potential contradictions, highlighting the absurdity of deducing anything from a false premise.
  • #1
Ssnow
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TL;DR Summary
Curiosity question on the infinite product ##2\cdot 2\cdot 2\cdots ##
Let us consider the infinite products ## p_{n}\,=\, 2\cdot 2\cdot 2 \cdot 2 \cdots 2 \,=\, 2^n## with ##n=1,\ldots ## . Clearly ##p_{n}\rightarrow +\infty## as ##n\rightarrow +\infty##. But if I put the infinity case ## 2\cdot 2\cdot 2 \cdot 2 \cdots \,=\, x## I have ##2\cdot x =x ## so ##x=0##. It is obvious I cannot put ##x=2\cdot 2\cdot 2 \cdot 2 \cdots ## and to try to seach the limit because the product diverges but has this "strange" algebraically formal result a conceptual reason to be (for example it is linked to the way to do the products ?) or it is only wrong and stop here ?
Thank you,
Ssnow
 
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  • #2
Ssnow said:
Summary:: Curiosity question on the infinite product ##2\cdot 2\cdot 2\cdots ##

Let us consider the infinite products ## p_{n}\,=\, 2\cdot 2\cdot 2 \cdot 2 \cdots 2 \,=\, 2^n## with ##n=1,\ldots ## . Clearly ##p_{n}\rightarrow +\infty## as ##n\rightarrow +\infty##. But if I put the infinity case ## 2\cdot 2\cdot 2 \cdot 2 \cdots \,=\, x## I have ##2\cdot x =x ## so ##x=0##.
Or ##x=\infty##. If you rule out ##\infty## here, then you are biasing the result.
 
  • #3
@FactChecker thanks, sure ##x=0 \vee x=\infty##. I ask for the absurd solution ##x=0## ...
Ssnow
 
  • #4
This is a pretty classic thing where you can create fake math. The real issue is that you started off by assuming a limit exists. If there is a limit and it is L (L is a real number), then 2L=L so L=0. But this assumes the limit exists to begin with, which obviously it does not.

You can get more obvious contradictions. 1+1+1+..., If it has a limit of L then 1+L= L so 1=0.
 
  • #5
@Office_Shredder I think ##1+L=L## imply that ##0L=-1## that is impossible!
In any case from something of false you can deduce everything ... :biggrin:
Thank you!
Ssnow
 

Related to Curiosity on this infinite product

1. What is "Curiosity on this infinite product"?

"Curiosity on this infinite product" refers to the concept of exploring and studying the vast and limitless possibilities of the universe and its components.

2. Why is curiosity important in science?

Curiosity is important in science because it drives scientists to ask questions, seek answers, and ultimately make new discoveries. It is the foundation of scientific inquiry and the key to expanding our knowledge and understanding of the world.

3. How does curiosity lead to innovation?

Curiosity leads to innovation by inspiring scientists to think outside the box and explore new ideas and possibilities. It encourages them to take risks and challenge traditional ways of thinking, ultimately leading to groundbreaking discoveries and advancements in various fields of science.

4. Can curiosity be taught or is it innate?

Both nature and nurture play a role in curiosity. Some people may have a natural inclination towards curiosity, but it can also be nurtured and developed through exposure to new experiences, encouragement, and a supportive learning environment.

5. How can we cultivate curiosity in ourselves and others?

We can cultivate curiosity by actively seeking out new information, asking questions, and being open-minded. It is also important to encourage and support curiosity in others, especially in young children, by providing them with opportunities for exploration and discovery.

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