Proving Natural Number e: Exponential Value Explained

In summary: The attempted manipulation was for a path running along the y-axis. The correct path to take is a sort of compromise between the two, as in this prescription:\displaystyle\lim_{{n_1}\rightarrow +\infty} {\displaystyle\lim_{{n_2}\rightarrow +\infty} {(1+\frac{1}{n_1})(1+\frac{1}{n_2})^{n_1}}}This prescription takes the limit along the y-axis, and then takes the limit along a diagonal line, and ends up with the value e. This is not something that can be seen by looking at the algebra of the formulas. It is something that must be discovered
  • #1
henrywang
14
0
For proving the natural number, e.
(1+1/n)^n As n approches infinite, (1+1/n)^n ----> e
However, wouldn't it become one as n becomes infinite?
(1+1/n)^n=(1+0)^n=1
Could anyone explain this to me please!
 
Mathematics news on Phys.org
  • #2
If you just naively substitute ##n=\infty##, then you would get

[tex](1+0)^\infty = 1^\infty[/tex]

which is an indeterminate form. Surely, things like ##1^2## and ##1^{100}## equal ##1##. But if the exponent is infinity, then it's an indeterminate form.
 
  • Like
Likes 1 person
  • #3
[tex](1+\frac{1}{n})^n = 1 + n\frac{1}{n} +\frac{ n(n-1)}{2!} \frac{1}{n^2} + ... \frac{1}{n^n}[/tex]

This does not ->1 as n becomes infinite.
 
Last edited:
  • Like
Likes 1 person
  • #4
henrywang said:
For proving the natural number, e.
(1+1/n)^n As n approches infinite, (1+1/n)^n ----> e
However, wouldn't it become one as n becomes infinite?
(1+1/n)^n=(1+0)^n=1
Could anyone explain this to me please!

The manipulation here takes a limit involving one variable and turns it into what amounts to a pair of nested limits involving two variables. The original formula was:

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

The attempted manipulation was to:

[itex]
\displaystyle\lim_{{n_1}\rightarrow +\infty} {\displaystyle\lim_{{n_2}\rightarrow +\infty} {(1+\frac{1}{n_2})^{n_1}}}
[/itex]

which is distinct from:

[itex]
\displaystyle\lim_{{n_2}\rightarrow +\infty} {\displaystyle\lim_{{n_1}\rightarrow +\infty} {(1+\frac{1}{n_2})^{n_1}}}
[/itex]

Those two nested limits give different results. The first one evaluates to 1. The second one does not exist at all. Limits do not always commute.

One way of seeing this is to imagine the formula [itex]{(1+\frac{1}{x})^y}[/itex] as a function of two variables. It is defined everywhere in the first quadrant of the coordinate plane for x and y. You can imagine extending the definition to infinite x and infinite y by tracing a path going off into the distance and looking at the limit of the function values along that path. It then turns out that the limit you get depends on the path you take.

The prescription in the original formulation was for a path running exactly on the 45° line through the middle of the first quadrant.
 
  • Like
Likes 2 people
  • #5


Thank you for your question. The value of e is indeed a fascinating and important concept in mathematics and science. Let me provide some explanation to help clarify the concept for you.

Firstly, it is important to understand that the expression (1+1/n)^n is an approximation of the value of e, not the exact value. As n approaches infinity, the expression becomes closer and closer to the actual value of e, but it will never be exactly equal to e. This is because the expression is a limit, meaning it is a mathematical concept used to describe how a function behaves as its input approaches a certain value, in this case, infinity.

Now, to address your question about the expression becoming 1 as n approaches infinity, this is actually not the case. The expression does indeed become closer and closer to 1 as n approaches infinity, but it will never actually equal 1. This is because as n approaches infinity, the value of 1/n approaches 0, but it will never actually be 0. Therefore, the expression will never simply be 1.

To further understand this concept, let's look at the graph of the function (1+1/n)^n. As you can see in the graph, as n approaches infinity, the function approaches the value of e, but it never actually reaches it. The function will continue to get closer and closer to e, but it will never cross the line and reach the exact value of e.

In conclusion, the expression (1+1/n)^n is a limit that approximates the value of e, but it will never be exactly equal to it. As n approaches infinity, the expression gets closer and closer to the value of e, but it will never become 1 or any other value. I hope this explanation helps clarify the concept for you.
 

1. What is the natural number e?

The natural number e, also known as Euler's number, is a mathematical constant that is approximately equal to 2.71828. It is an irrational number, meaning it cannot be expressed as a simple fraction, and it is the base of the natural logarithm.

2. How is the value of e calculated?

The value of e can be calculated in several ways, including using the infinite series 1 + 1/1! + 1/2! + 1/3! + ..., where n! represents the factorial of n. It can also be calculated using the limit definition of the exponential function, where the limit of (1 + 1/n)^n as n approaches infinity is equal to e.

3. Why is e considered a fundamental constant in mathematics?

E is considered a fundamental constant in mathematics because it appears in many important mathematical formulas and equations, such as compound interest, exponential growth and decay, and the normal distribution. It also has connections to other fundamental mathematical constants, such as pi and the imaginary number i.

4. How is e used in real-world applications?

The natural number e has numerous real-world applications, particularly in the fields of finance, economics, and science. It is used in compound interest calculations, population growth models, and in the study of radioactive decay. It also plays a role in signal processing, electrical engineering, and fluid dynamics.

5. Can the value of e be proved mathematically?

Yes, the value of e can be proved mathematically using various techniques, such as the ones mentioned in question 2. Additionally, there are many proofs of the irrationality and transcendence of e, which further support its fundamental role in mathematics.

Similar threads

Replies
3
Views
530
  • General Math
Replies
3
Views
982
Replies
12
Views
2K
Replies
4
Views
253
Replies
4
Views
1K
  • General Math
Replies
1
Views
674
  • General Math
Replies
4
Views
1K
Replies
1
Views
733
Replies
6
Views
909
Back
Top