Can the Value of e Be Found Using the Taylor Series?

[johndoe]

Why does
$$\lim(1+\frac{1}{x})^x = e$$
$$x->\infty$$

 

[Pere Callahan]

This is one possible definition of the number e.

What is your definition of e?

 

[johndoe]

This is one possible definition of the number e.

What is your definition of e?

O so how do i prove this ?

 

[rock.freak667]

Let

$$L=\lim_{x\rightarrow \infty} (1+\frac{1}{x})^x$$

$$ln L =ln \lim_{x\rightarrow \infty} (1+\frac{1}{x})^x$$

$$ln L =\lim_{x\rightarrow \infty} xln (1+\frac{1}{x})$$

Then use L'Hopital's Rule twice.

 

[Hurkyl]

O so how do i prove this ?

If you don't have a definition of e, then you can't.

 

[Pere Callahan]

O so how do i prove this ?

How do you prove what?

 

[ice109]

How do you prove what?

yea how do you prove a definition? maybe he means how do you consistency?

 

[HallsofIvy]

Unfortunately, johndoe hasn't gotten back to us to answer the question about what definition of e he is using.

Yes, many texts define e by that limit. If that is the definition, then no "proof" is required.

Others, however, treat the derivative of f(x)= a^x this way:
$$f(x+ h)= a^{x+ h}= a^x a^h$$
$$f(x+ h)- f(x)= a^xa^h- a^x= a^x(a^h- 1)$$
$$\frac{f(x+h)- f(x)}{h}= a^x\frac{a^h- 1}{h}$$
Which, after you have shown that
$$\lim_{h\rightarrow 0}\frac{a^h-1}{h}$$
exists, shows that the derivative of a^x is just a constant times a^x.
"e" is then defined as the value of a such that that constant is 1:
$$\lim_{h\rightarrow 0}\frac{e^h- 1}{h}= 1$$

We can then argue (roughly, but it can be made rigorous) that if, for h close to 0, (e^h-1)/h is close to 1, e^h- 1 is close to h and so e^h is close to h+ 1. Then, finally, e is close to (h+1)^1/h. As h goes to 0, 1/h goes to infinity. letting x= 1/h, (h+1)^1/h becomes (1/x+ 1)^x so
$$\lim_{x\rightarrow \infty}(1+ \frac{1}{x})^x= e$$

Here's a third definition of "e":
It has become more common in Calculus texts to work "the other way". That is, define ln(x) by
[tex]ln(x)= \int_1^x \frac{1}{t}dt[/itex]
From that we can prove all the properties of ln(x) including the fact that it is a "one-to-one" and "onto" function from the set of positive real numbers to the set of all real numbers- and so has an inverse function. Define "exp(x)" to be the inverse function . Then we define "e" to be exp(1) (one can show that exp(x) is, in fact, e to the x power).

Now, suppose $lim_{x\rightarrow \infty} (1+ 1/x)^x$ equals some number a. Taking the logarithm, x ln(1+ 1/x)must have limit ln(a). Letting y= 1/x, that means that the limit, as y goes to 0, of ln(1+y)/y must be ln(a). But that is of the form "0/0" so we can apply L'hopital's rule: the limit is the same as the limit of 1/(y+1) which is obviously 1. That is, we must have ln(a)= 1 or a= e.

 

[johndoe]

Unfortunately, johndoe hasn't gotten back to us to answer the question about what definition of e he is using.

Yes, many texts define e by that limit. If that is the definition, then no "proof" is required.

Others, however, treat the derivative of f(x)= a^x this way:
$$f(x+ h)= a^{x+ h}= a^x a^h$$
$$f(x+ h)- f(x)= a^xa^h- a^x= a^x(a^h- 1)$$
$$\frac{f(x+h)- f(x)}{h}= a^x\frac{a^h- 1}{h}$$
Which, after you have shown that
$$\lim_{h\rightarrow 0}\frac{a^h-1}{h}$$
exists, shows that the derivative of a^x is just a constant times a^x.
"e" is then defined as the value of a such that that constant is 1:
$$\lim_{h\rightarrow 0}\frac{e^h- 1}{h}= 1$$

We can then argue (roughly, but it can be made rigorous) that if, for h close to 0, (e^h-1)/h is close to 1, e^h- 1 is close to h and so e^h is close to h+ 1. Then, finally, e is close to (h+1)^1/h. As h goes to 0, 1/h goes to infinity. letting x= 1/h, (h+1)^1/h becomes (1/x+ 1)^x so
$$\lim_{x\rightarrow \infty}(1+ \frac{1}{x})^x= e$$

Here's a third definition of "e":
It has become more common in Calculus texts to work "the other way". That is, define ln(x) by
[tex]ln(x)= \int_1^x \frac{1}{t}dt[/itex]
From that we can prove all the properties of ln(x) including the fact that it is a "one-to-one" and "onto" function from the set of positive real numbers to the set of all real numbers- and so has an inverse function. Define "exp(x)" to be the inverse function . Then we define "e" to be exp(1) (one can show that exp(x) is, in fact, e to the x power).

Now, suppose $lim_{x\rightarrow \infty} (1+ 1/x)^x$ equals some number a. Taking the logarithm, x ln(1+ 1/x)must have limit ln(a). Letting y= 1/x, that means that the limit, as y goes to 0, of ln(1+y)/y must be ln(a). But that is of the form "0/0" so we can apply L'hopital's rule: the limit is the same as the limit of 1/(y+1) which is obviously 1. That is, we must have ln(a)= 1 or a= e.

Ok I see. I didn't know that definition before until I reach a problem requiring me to find that limit.

 

[prasannaworld]

I believe that the definition of e is:

e is such that when: y=e^x
dy/dx = e^x

Hence via the Taylor Series, the actual value of e can be found

right?