Calculating the Limit of (1+1/n)^n as n Approaches Infinity

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  • Thread starter Thread starter Matt Jacques
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Discussion Overview

The discussion revolves around the computation and understanding of the mathematical constant "e," particularly through the limit of the expression (1 + 1/n)^n as n approaches infinity. The scope includes theoretical exploration and mathematical reasoning related to the definition and properties of "e."

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant inquires about how "e" was computed, prompting a discussion on various methods.
  • Another participant suggests that there are multiple ways to compute "e" and provides a link to a resource for further exploration.
  • A different participant defines "e" in terms of an integral, stating that it is the number such that the integral of 1/x from 1 to e equals 1, and discusses how this can be used to approximate "e."
  • This same participant also mentions using the Taylor series for e^x, indicating that plugging in x = 1 leads to the limit of the series representing "e."
  • A final post presents the mathematical limit \lim_{n\to\infty} (1+\frac{1}{n})^n, suggesting a direct inquiry into the limit itself.

Areas of Agreement / Disagreement

Participants present various methods and definitions related to "e," but there is no consensus on a singular approach or resolution to the inquiry about its computation.

Contextual Notes

Some methods mentioned depend on specific mathematical definitions and assumptions, such as the properties of integrals and series, which may not be universally accepted or understood in the same way by all participants.

Matt Jacques
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How was "e" computed?

The title says it all.
 
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the number e is by definition the number e such that the integral of 1/x from 1 to e equals 1. thus any method of approximating integrals allows you to approximate e. i.e. if yuo show that the integral of 1/x from 1 to 2.7 is less than 1 you have shown that e is greater than 2.7


another nice way is to use the taylor series for e^x, and plug in x =1, which gives e = the limit of the series 1 + 1/2 + 1/6 + 1/24 + 1/120 + ...+ 1/n!+...


My freshman calc prof gave as an exercise to prove this way that e starts out as 2.718281828... but i never completed it. i could not believe in those days that math took that much work!
 
calculate [tex]\lim_{n\to\infty} (1+\frac{1}{n})^n[/tex]
 

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