Defining and Evaluating the Limit of e: A Closer Look

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In summary, there are various ways to define the number e and its value can be proven using different definitions. One possible definition is e = lim n->infinity (1+1/n)^n. Another definition is e = sum n->infinity 1/n!. It can also be defined as the unique function f(x) such that f(1) = 1 and f'(x) = f(x) for all real numbers x. Ultimately, the value of e can be derived using different definitions and proofs, making it a versatile and important number in mathematics.
  • #1
cscott
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How to you evaluate the expression for e (the limit) ? I don't see how you could do this unless you do it numerically since e is irrational:confused:
 
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  • #2
What do you mean by "evaluate"?

(And you realize the square root of 2 is irrational, right)
 
  • #3
What limit are you talking about? There exist an infinite number of functions or sequences that have limit e. Of course, how you evaluate a limit has nothing to do with whether the limit is rational or irrational.
 
  • #4
Ok, I'll just ask this:

[tex]\lim_{x\rightarrow\infty} (1 + \frac{1}{n})^n[/tex]

How would I find that limit?
 
  • #5
That limit's easy: it's (1 + 1/n)^n. :smile:

That limit, once you fix the typo, will be equal to e. How you prove it depends on what you use as your definition of e. (Some people use that limit as the definition of e, so it's a rather trivial proof!)
 
  • #6
The only way I've ever been able to prove it (without really thinking about the topic - I'm sure there are other proofs) is to use logarithms and L'H[tex]\hat{o}[/tex]pital's rule. I'll start it off and you can fill in the rest.

[tex]y = (1 + \frac{1}{n})^n[/tex]

[tex]ln(y) = ln(1 + \frac{1}{n})^n[/tex]

[tex]ln(y) = n \cdot ln(1 + \frac{1}{n})[/tex]

[tex]\lim_{n\rightarrow \infty} ln(y) = \lim_{n\rightarrow \infty} n \cdot ln(1 + \frac{1}{n})[/tex]

Now the right side is [tex] \infty \cdot 0[/tex] so you can apply L'H[tex]\hat{o}[/tex]pital's rule.
 
  • #7
BSMS,

But you're using the value of e to derive the value of e with the natural logarithm - that's somewhat circular.
 
  • #8
If you never use the value of e, you cannot possibly prove that the limit is e. :-p
 
  • #9
You can prove that the limit is some number, and that it's, say, less than 3. Or you can compute it numerically by plugging in larger and larger values of n to get better and better approximations. But, as has been mentioned, you need a definition of e to show it equals e. One thing you could do is show different definitions are equivalent. For example, prove

[tex]\lim_{n\rightarrow \infty} (1+\frac{1}{n})^n = \sum_{n=0}^{\infty} \frac{1}{n!}[/tex]
 
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  • #10
So we can use,

[tex]\lim_{h\rightarrow0} \frac{e^h - 1}{h} = 1[/tex]

as our definition and with it we can show the limit in my above post is equal to e?
 
  • #11
cscott said:
So we can use,

[tex]\lim_{h\rightarrow0} \frac{e^h - 1}{h} = 1[/tex]

as our definition and with it we can show the limit in my above post is equal to e?
yes

we can make any number of definitions for e and prove they are equivelent. For each definition to be valid we should prove that there exist at least one number satisfying the definition (existence), and that there do not exist more than one number satisfying the definition (uniqueness).

Your definition is my preferred one, but a possible problem is the function a^x must be defined (again any number of definitions are possible).

I personally patch this up this way.

Theorem:there exist a function f:R->R such that for all real numbers x,y
a) f(x)*f(y)=f(x+y)
b) lim x(real)->0 [f(x)-1]/x=1
Theorem: if f and g are two functions as above f=g

Definition: if f is a function as above
e:=f(1)

Several potential definitions of e are
e=lim n(natural)->infinity (1+1/n)^n
e=sum n(natural) 1/n!
1=lim x(real)->0 (e^h-1)/h (having defined a^x)
if f'(x)=f(x) f(0)=1 e=f(1)
log(e)=1 (having defined log(x))

There are other likely more interesting possibilities
 

FAQ: Defining and Evaluating the Limit of e: A Closer Look

What is the mathematical definition of "e"?

The mathematical constant "e" is approximately equal to 2.71828 and is the base of the natural logarithm. It is an irrational number, meaning it cannot be expressed as a simple fraction, and has infinite decimal digits without any repeating pattern.

How is "e" used in mathematical expressions?

The constant "e" is often used in mathematical expressions involving exponential growth or decay. It is also used in calculus to represent the slope of the tangent line to the graph of the natural exponential function y=e^x at any point.

What is the significance of "e" in real-world applications?

The constant "e" has numerous applications in fields such as finance, physics, and biology. It is commonly used to model continuous growth and decay, such as in population growth or compound interest. It also plays a role in determining probabilities and in solving differential equations.

How is "e" related to other mathematical constants?

"e" is closely related to other important mathematical constants such as pi and the golden ratio. For example, e^(pi*i) = -1, where i is the imaginary unit. The golden ratio can also be expressed in terms of "e" as (1+sqrt(5))/2 = e^(ln(1+sqrt(5)))

Can the value of "e" be approximated?

Yes, the value of "e" can be approximated using various methods such as the Maclaurin series or continued fraction expansion. One common approximation is 2.71828, but more precise calculations can be made by using more terms in the series or expansion.

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