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Homework Help: A question regarding the definition of e

  1. Dec 15, 2014 #1
    1. The problem statement, all variables and given/known data
    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)?

    2. Relevant equations

    3. The attempt at a solution
    I see no clue as to why is this so.
  2. jcsd
  3. Dec 15, 2014 #2


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    Science Advisor

    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, ..."
  4. Dec 15, 2014 #3


    Staff: Mentor

    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
  5. Dec 15, 2014 #4


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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.
  6. Dec 15, 2014 #5


    Staff: Mentor

    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.
  7. Dec 15, 2014 #6

    Ray Vickson

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    Homework Helper

    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.
  8. Dec 15, 2014 #7
    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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