A question regarding the definition of e

In summary, the variable "n" is used in the definition of ##e## due to its connection to positive integers.
  • #1
SafiBTA
10
0

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.
 
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  • #2
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
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
To get a bit more pedantic, the "limit" is the single, fixed numeric value (if one exists) that the sequence approaches.
 
  • #5
SafiBTA said:
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
SafiBTA said:

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
jedishrfu said:
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!
 

What is the definition of e?

The mathematical constant e, also known as Euler's number, is approximately equal to 2.71828. It is the base of the natural logarithm and is commonly used in calculus and other areas of mathematics.

How is e calculated?

Euler's number can be calculated by taking the limit of (1 + 1/n)^n as n approaches infinity. This value is approximately equal to 2.71828, but can also be calculated to higher precision using various mathematical methods.

What is the significance of e?

Euler's number has many applications in mathematics, including in calculus, differential equations, and growth and decay problems. It is also used in various fields such as finance, physics, and engineering.

How is e related to compound interest?

The value of e is closely related to the concept of compound interest, which is interest that is calculated not only on the initial principal, but also on the accumulated interest of previous periods. The larger the number of compounding periods, the closer the interest rate will be to e.

What is the difference between e and pi?

Euler's number and pi (π) are both important mathematical constants, but they serve different purposes. While e is used in exponential and growth functions, pi is used in circular and trigonometric functions. Additionally, e is an irrational number, meaning it cannot be expressed as a fraction, while pi is a transcendental number, meaning it cannot be the root of any algebraic equation with rational coefficients.

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