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A question regarding the definition of e

  • #1
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Homework Statement


In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have anything to do with positive integers (which we normally use 'n' for)?

Homework Equations


##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##

The Attempt at a Solution


I see no clue as to why is this so.
 

Answers and Replies

  • #2
jbriggs444
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The n in that expression for e is not a free variable. It is a bound variable. It makes no sense to talk about the formula holding for some particular value of n. The n in the formula has a limited scope and is not defined at all outside of that scope.

Have you ever encountered a formal definition for "limit"? In your coursework, have you encountered statements such as "there exists an x in R such that for all n in N, ..."
 
  • #3
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The limit describes a sequence of numbers that as n increases becomes closer and closer to e.

n=1 and the expr=2
n=2 and the expr=2.2
n=3 and the expr=2.37
n=4 and the expr=2.44
...
n approaches infinity expr approaches e
 
  • #4
jbriggs444
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To get a bit more pedantic, the "limit" is the single, fixed numeric value (if one exists) that the sequence approaches.
 
  • #5
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In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have anything to do with positive integers (which we normally use 'n' for)?
By "formula" I presume you mean this expression: (1 + 1/n)n. Clearly it is not defined for n = 0. The "notation," as you called it, is a limit, and the expression whose limit is being taken has different values for different values of n (integer values). As n gets larger, the value of the expression (1 + 1/n)n gets closer to the number e.
 
  • #6
Ray Vickson
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Homework Statement


In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have anything to do with positive integers (which we normally use 'n' for)?

Homework Equations


##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##

The Attempt at a Solution


I see no clue as to why is this so.
It is true that
[tex] e = \lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x[/tex]
but that is a theorem, rather than a definition.

The point is that concepts/objects need to be defined before they are used (in a strictly logical approach, at least), so that although one can define and understand ##a^n## for real ##a## and integer (or even rational) ##n##, ##a^x## for real ##x## is trickier and more involved to get at. Sometimes, the definitions related to ##a^x## proceed through the use of the number ##e## itself, where it is understood that this ##e## is the one obtained by the usual integer limit.
 
  • #7
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The limit describes a sequence of numbers that as n increases becomes closer and closer to e.

n=1 and the expr=2
n=2 and the expr=2.2
n=3 and the expr=2.37
n=4 and the expr=2.44
...
n approaches infinity expr approaches
Suppose there is a bank A which gives 100%interest per annum , and bank Bgives 100 percent but compounded twice in a year,and bank C gives 100 percent per annum but compounded thrice(i,e every for months ) and so on........
Now Dollar 1 deposited in each of these banks will be 2,(1plus1/2)power2,(1plus1/3) power3 and so on .
Ultimately in this way a hypothetical bank which compounds interest every moment will give you back Dollars e!
 

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