Why (1+1/n)^n goes to e as n goes to infinite?

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Homework Help Overview

The discussion revolves around the mathematical expression (1 + 1/n)^n and its limit as n approaches infinity, specifically relating it to the number e. Participants are exploring how to justify this limit and the definitions involved.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants suggest using logarithms and l'Hopital's rule as a method to justify the limit. Others mention testing large values of n to observe the convergence towards e. There are questions about the definition of e and its implications in the context of the limit.

Discussion Status

The discussion is active with various approaches being considered. Some participants provide insights into definitions and properties of e, while others express curiosity about the convergence of the sequence. There is no explicit consensus yet on the best method to demonstrate the limit.

Contextual Notes

Participants are discussing different definitions of e and the nature of the limit, indicating a potential lack of clarity on foundational concepts. There is also a mention of the importance of the number e in mathematics, which may influence the discussion's direction.

Teachme
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I uploaded a picture of my question. I am just wondering how to justify how (1 + 1/n)^n goes to e as n goes to ∞? How do you show this?
Thanks!
 

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You can if you take the log and then use l'Hopital's rule. On the other hand it might just be a definition of 'e'.
 
You could alose plug in some very large values of n and see that it indeed approaches e (I was bored and did this myself a few days ago). This is the definition of e.
 
nevermind.
 
How, exactly are you defining "e"? In many texts, e is defined by that limit.
 
It's more instructive to show that the sequence converges. Its limit is denoted by 'e' and is one of the most important real numbers in mathematics. Another interesting proof would be to show that the number, namely the limit, is irrational. Then transcendental.
 

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