Proving lim (1-1/n)^n=1/e as n Approaches Infinity

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In summary: Or you could have started with the fact that the derivative of e^x is e^x and, using the chain rule, that the derivative of e^u is u'.e^u. Then the derivative of e^(1/n) is \frac{-1}{n^2}e^{1/n}. Since the derivative of e^x is e^x itself, \left(\frac{-1}{n^2}e^{1/n}\right)^2= e^{1/n}so e^{1/n}= \frac{-1}{n^2}e^{1/n} Now, the derivative of e^(1/n)
  • #1
MooCow
6
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Show that:

lim (1-1/n)^n=1/e
n->infinity

I don't really know where to begin...
 
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  • #2
MooCow said:
Show that:

lim (1-1/n)^n=1/e
n->infinity

I don't really know where to begin...

Well, I guess it depends what you are allowed to start with. You could try to use the limit deffinition which is used to define interest which is compounded continuously. Another thought would be to do a series expansion. Does your book give you a deffinition of e?
 
  • #3
first what can you start with for e?
What is given as the definition of e?
by elementaty algebra
(1-1/n)^n=1/(1+1/(n-1))^n
lim (1+1/(n-1))^n=e
n->infinity
 
  • #4
Try taking the logarithm of both sides, then applying L'hopital's rule.
 
  • #5
nicksauce said:
Try taking the logarithm of both sides, then applying L'hopital's rule.

That won't work because we are trying to establish the equality, thus we can not assume it to be true. Something similar works however.
let lim denote the limit n->infinity n a natural number

we wish to show
lim (1-1/n)^n=1/e
consider
log(lim (1-1/n)^n)=lim log((1-1/n)^n)
supose we know log is continuous at 1, then
log(lim (1-1/n)^n)=lim log((1-1/n)^n
perhaps we know log(x^y)=y*log(x), thus
log(lim (1-1/n)^n)=lim n*log((1-1/n)^n)
log(lim (1-1/n)^n)=lim -log(1-1/n)/(-1/n)
log(1)=0 so
log(lim (1-1/n)^n)=lim -[log(1-1/n)-log(1)]/(-1/n)
log(lim (1-1/n)^n)=-log'(1)
log(lim (1-1/n)^n)=-1
taking antilogs of both sides
lim (1-1/n)^n=exp(-1)=1/e
as desired
we do not need L'hopitals rule since we have the definition of the derivative
 
  • #6
lurflurf said:
That won't work because we are trying to establish the equality, thus we can not assume it to be true. Something similar works however.
let lim denote the limit n->infinity n a natural number

we wish to show
lim (1-1/n)^n=1/e
consider
log(lim (1-1/n)^n)=lim log((1-1/n)^n)
supose we know log is continuous at 1, then
log(lim (1-1/n)^n)=lim log((1-1/n)^n
perhaps we know log(x^y)=y*log(x), thus
log(lim (1-1/n)^n)=lim n*log((1-1/n)^n)
log(lim (1-1/n)^n)=lim -log(1-1/n)/(-1/n)
log(1)=0 so
log(lim (1-1/n)^n)=lim -[log(1-1/n)-log(1)]/(-1/n)
log(lim (1-1/n)^n)=-log'(1)
log(lim (1-1/n)^n)=-1
taking antilogs of both sides
lim (1-1/n)^n=exp(-1)=1/e
as desired
we do not need L'hopitals rule since we have the definition of the derivative

Hi, in your proof above, where do we use the fact that "log is continuous at 1"? For derivative definition at the end, or, for interchanging log and limit?

And one more question if you don't mind.

" Given f(n) a monotonically increasing sequence going to infinity as n goes to infinity.
Is it always true that lim(1 - 1/f(n))^f(n) = 1/e? "

Thanks a lot for your answer above, and any answer you may have for my two questions.
 
  • #7
" Given f(n) a monotonically increasing sequence going to infinity as n goes to infinity.
Is it always true that lim(1 - 1/f(n))^f(n) = 1/e? "
Yes. In this is the substitution principle
if lim_{x->a} f(x)=L
and lim_{x->a}g(x)=a
then lim_{x->a}f(g(x))=L
When the variable is n often the limit is taken over integers and in general the domain of the limit variable matters.

Hi, in your proof above, where do we use the fact that "log is continuous at 1"? For derivative definition at the end, or, for interchanging log and limit?
For both. "log is continuous at 1" is one of several ways of making sure log is not a messed up function, that is nowhere continuous.

To define log (for real numbers) it is sufficient to require
1) for all x,y real log(xy)=log(x)+log(y)
2) log'(1)=1 (and thus log'(1) exists)
or
2*)log(e)=1 and there is at least one point where log is continuous
Condition 2) serves two purposes
We assure a nice function and specify the natural logarithm.
 
  • #8
MooCow said:
Show that:

lim (1-1/n)^n=1/e
n->infinity

I don't really know where to begin...
One serious problem you have with this is that it isn't true!
What is true is that
[tex]\lim_{n\to\infty}\left(1- \frac{1}{n}\right)^n= e[/tex]
not "1/e".

How you show that depends upon exactly how you have defined "e". In fact, in a recent thread, the question was asked how to show that the derivative of [itex]e^x[/itex] is [itex]e^x[/itex] where the formula you have was given as a definition for e. What we can do here is the reverse of what I did in that thread.

Now, what definition of "e" are you using? Simplest, if you are not going to use that formula itself, is that ln(e)= 1 where ln(x) is defined by
[tex]ln(x)= \int_1^x \frac{1}{t}dt[/tex].

You can show from that that "ln(x)" and "[itex]e^x[/itex]" are inverse functions. And, since the derivative of ln(x) is 1/x, it follows that the derivative of [itex]e^x[/itex] is [itex]e^x[/itex] itself. Or, if you prefer to define [itex]e^x[/itex] by the fact that its derivative is itself, you can start from that fact.

Now, of course, the derivative of [itex]e^x[/itex] is given by
[tex]\lim_{h\to 0}\frac{e^{x+h}- e^x}{h}= \lim_{h\to 0}\frac{e^xe^h- e^x}{h}= e^x\lim_{h\to 0}\frac{e^h- 1}{h}[/tex]

And, since the derivative is [itex]e^x[/itex] that means we must have
[tex]\lim_{h\to 0}\frac{e^h- 1}{h}= 1[/tex].

That, now, means that for h very close to 0,
[tex]\frac{e^h- 1}{h}\approx 1[/tex]
so that
[tex]e^h- 1\approx h[/tex]
For any h> 0, there exist an integer n such that n> 1/h. With such an n, we have
[tex]e^{1/n}- 1\approx \frac{1}{n}[/tex]
so that
[tex]e^{1/n}\approx 1+ \frac{1}{n}[/tex]
[tex]e\approx \left(1+ \frac{1}{n}\right)^n[/tex]
Now taking the limit as n goes to infinity is the same as taking the limit as h goes to 0 so the "approximation" becomes exact:
[tex]e= \lim_{n\to\infty} \left(1+ \frac{1}{n}\right)^n[/tex]
 
  • #9
Halls:
You are wrong.
There is a - in the parenthesis, not a +.
 
  • #10
Unless i am serioysly deceived the thread is 2 years old btw ... And this seems to be an excersice for first year or indroductory course to analysis, so couldn't possible have to do with continuity, derivatives and Lhopital rule. Post #3 also posted 2 years ago seem to do the proper job in answering.
 
  • #11
Details, details :))

Anyways, Halls' post is very good.
 
  • #12
Except for that minor detail with the negative!

Ah, well, the operation was successful even though the patient died.
 
  • #13
HallsofIvy said:
Except for that minor detail with the negative!

Ah, well, the operation was successful even though the patient died.
:smile:
 
  • #14
and now that i see again i am seriously deceived cause the thread is almost 3 years old...
 
  • #15
Hmm... it gave me a thought. If

[tex]\left(1+\frac{1}{n}\right)^n = e[/tex]

then what is

[tex]\left(1+\frac{1}{-n}\right)^{-n}[/tex]

?

As n approaches positive infinity, of course.
 
  • #16
Simplify and it becomes clear: [tex]lim_{n \to \infty} \frac{1}{(1 - \frac{1}{n})^n} = \frac{1}{\frac{1}{e}} = e[/tex]. You can do this because the negative n in the fraction turns it into a negative, the negative n in the exponent turns it into division, and the limit of a fraction is equal to the fraction of the numerator's and denominator's limit, when the denominator isn't 0.
 
  • #17
But wolfram alpha tells me the limit should be e.
 
  • #18
See it again. I made a dumb mistake, then edited.
 

Related to Proving lim (1-1/n)^n=1/e as n Approaches Infinity

1. What is the significance of proving lim (1-1/n)^n=1/e as n Approaches Infinity?

The limit (1-1/n)^n=1/e is an important concept in calculus and is often used in applications such as compound interest, population growth, and radioactive decay. It also has connections to the natural logarithm and the exponential function.

2. How do you prove lim (1-1/n)^n=1/e as n Approaches Infinity?

To prove this limit, we can use the definition of the limit and the properties of limits. First, we can rewrite the expression as lim (1-1/n)^n=lim (1-1/n)^n * 1=lim (1-1/n)^n * (n/n) = lim [(1-1/n)^n * (n/n)] = lim [(1-1/n)^n * (1/(1/n))] = lim [(1-1/n)^(n*n)] = lim (1-1/n)^n = lim (1-1/n)^n * lim (n/n) = lim (1-1/n)^n * lim (n/n) * lim (n/n) = lim (1-1/n)^n * lim (n/n) * lim (1/n) = lim [(1-1/n)^n * (1/n)] = lim (1-1/n)^n * (1/n) = lim (1-1/n)^n * lim (1/n) * lim (1/n) * lim (1/n) = lim (1-1/n)^n * lim (1/n) * lim (1/n) * lim (1/n) = lim (1-1/n)^n * lim (1/n) = 1 * 1 = 1/e. Therefore, lim (1-1/n)^n=1/e as n Approaches Infinity.

3. Can you explain the intuition behind lim (1-1/n)^n=1/e as n Approaches Infinity?

The limit (1-1/n)^n=1/e can be thought of as the continuous compounding of interest or growth over an infinite number of intervals. As n approaches infinity, the expression (1-1/n) gets closer and closer to 1, making the overall expression approach 1/e, which is the value of the exponential function at a rate of 1.

4. What is the connection between lim (1-1/n)^n=1/e and the natural logarithm?

The natural logarithm, ln(x), is defined as the inverse function of the exponential function, e^x. Therefore, the value of ln(e) is equal to 1. By proving lim (1-1/n)^n=1/e, we are essentially showing that the limit of (1-1/n)^n approaches the value of e. This reinforces the concept that the natural logarithm is the inverse of the exponential function.

5. Are there any real-world applications of lim (1-1/n)^n=1/e?

Yes, there are many real-world applications of this limit. For example, it can be used to model the growth of populations, the decay of radioactive materials, and the accumulation of interest in compound interest calculations. It is also used in finance, physics, and engineering to model various natural phenomena and make predictions.

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