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

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SUMMARY

The limit of the expression (1 + 1/n)^n approaches the mathematical constant e as n approaches infinity. This can be justified using logarithmic properties and l'Hôpital's rule. Additionally, the definition of e is often presented through this limit, emphasizing its significance in mathematics. Exploring the convergence of the sequence and proving that e is both irrational and transcendental are also valuable insights discussed.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with logarithmic functions
  • Knowledge of l'Hôpital's rule
  • Basic concepts of irrational and transcendental numbers
NEXT STEPS
  • Study the application of l'Hôpital's rule in calculus
  • Explore the properties of logarithms and their applications
  • Research the proof of the irrationality of e
  • Learn about transcendental numbers and their significance in mathematics
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Mathematics students, educators, and anyone interested in understanding the properties of the constant e and its implications in calculus and number theory.

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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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