Euler's number as a limit - How to compute it
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- čas přidán 12. 02. 2023
- Hi! I'm Mateo Patiño, and I record math and physics videos. Most of my content is based on problem walkthroughs and mini-lectures where I discuss a particular concept. Feel free to subscribe if you like the content or give me any feedback on things you'd like me to improve! 🍎
In this video, I will show you how to compute Euler's number as a limit of an exponential function. This limit is pretty common to see in calculus, and there are many ways to compute it. The method I use in this video is re-writing the exponential function as the derivative of natural logarithm evaluated at x = 1.
Don't hesitate to ask me any questions 😃
I like the video but isn't this circular reasoning? You cannot say that the limit (the one where delta x approaches zero) is the derivative of ln(x) if the value of the limit is needed to prove the derivative of ln(x) at x=1.
Absolutely, i thought he was trying to calculate the value of “e”.😂
@@adventoabdielnababan1952me too
Definitely it's a kind of circular proof.
Or even a trivial proof like this:
"Suppose that γ is equal to 'e'. So because the limit of the expression (1+1/n)^n is equal to γ and γ is equal to 'e', then the limit is also equal to 'e'.
The author of the video is genius!
While the author of this video certainly didn‘t elaborate that far, one can avoid circularity. For instance, you could define ln(x) as the antiderivative of 1/x.
@@phscience797
@phscience797
"you could define ln(x) as the antiderivative of 1/x"
No, you couldn't. You may not use any derivatives of exp(x) or ln(x) till you prove the Second Wonderful limit.
Teaching many concepts in 1 video. Thank you very much!
This is NOT just a way to define e. This is the ORIGINAL way.
Really liked your explanation✌️please do more videos on how natural logarithmic table was first made
Not to sound harsh, but you haven’t really shown anything in this video. You have shown that the limit of (1+1/n)^n as n approaches infinity is the number e. BUT THAT IS THE DEFINITION of e! How can you prove a definition? At the same time, you did not compute e, which you said you would do.
He showed us how to compute the limit in the standard way rather than using the usual form which is exp(Limit x tends to (..) f(x)/g(x)) for the 1^infinity form
Nicely explained, but it doesn't match the description of the video! The video stated how to calculate e, which of course immediately conjures the Taylor series expansion for e^x evaluated at 1. Nevertheless, I appreciated your proof that this limit approaches e. .... Here's a related question you might want to consider: how well does this formula approximate e when n is some value like 10 or 100 or 1000. How would you characterize the convergence?
An easy way to see that something is amiss is to use a logarithm to another base. Say use log base 10, then you’d prove that the limit equals 10, not e.
nga how do you fine ln(x) without knowing e
Your explanation is dammm good....teach me also in some video😅😅plz plz
Some people already commented, but it is good to make it clear:
this proof is invalid.
But it is also common. So this youtuber shouldn't feel bad at all.
i have doubt in my mind on this way of proving that this involves lo hopitals rule and taking log abe e itself , actually think there musty be algeraic way to arrive here which is based on more rigour, hope u helpme with this
Circular reasoning. You cannot use ln (log base e) trying to define e itself.
👍
Future MIT professor here
Well, he is young, so he can be whatever he wants to if he works hard for it.
There is only one way to define e -- e=limit (1+1/x)^x. Euler defined e as a Continuous Compounding of interest rates which is limit (1+i/x)^(tx)which iis equal e^(it). If you set i and t to 1 you get e. e is not an abstract concept but has real physical meaning. And he is trying to show that e = e
Didn't Bernoulli do that?
@@Memories_broken_ There are a lot of pretenders on the throne of been 1st. Euler is not one of them. But Euler was the one who named the constant in his own honour.😄😄😄😄
Even more interesting is that we are using the ln (base e) to prove that e equal to e
@@Grim_Reaper_from_Hell oh no Bernoulli wasn't a pretender. He did notice the number appearing while working with interests and compounded sums, he just didn't dwell into it.
@@Memories_broken_ Bernoulli is not the only one. If you look through the various books on the topic you will find a number of different names. Compounding was a popular topic back then and it is not surprising that a number of people derived the number before Euler and Bernoulli was one of them but was the Bernoulli 1st is not clear.
You are just 4wful. Have some decency.
Wait a minute. How do you know that the derivative of ln x is 1/x without knowing that the limit you started with is equal to e??
Indeed, the proof that shows that the derivative of lnx is one over x already assumes the limit equals e. I didn't know this when I recorded this video, but as others say, this feels like circular reasoning!
@@TheCalcSeries this doesn't "feel like" circular reasoning. This IS circular reasoning. But that's ok. Pretty common in proofs involving e.
Usually people make circular reasoning to prove the continuity or differentiability of e^x. Your video is NOT about those. You just make a little honest understandable confusion.
The limit in the video is the ORIGINAL definition of e. It gives the expression
e = 1/0! + 1/1! + 1/2! + ...
To see that, notice that for fixed n,
(1+1/n)^n
= sum_k C(n,k) (1/n)^k
= sum_k n!/(k!(n-k)!) (1/n)^k
= sum_k n(n-1)...(n-(k-1))/k! 1/n^k
= sum_k 1/k! n/n (n-1)/n ... (n-(k-1))/n
= sum_k 1/k! 1 (1-1/n) ... (1-(k-1)/n)
of course, k goes from 0 to n. For n big, each term 1-k/n approximates 1 and products of those terms also approxinate 1, so each term of the sum will be approximately 1/k!, which means
lim (1+1/n)ⁿ = sum_k 1/k!
Some particular examples:
n=1
(1+1/1)¹ = 1+1
n=2
(1+1/2)²
= 1+2(1/2)+(1/2)²
= 1+1+1/4
= 2.25
n=3
(1+1/3)³
= 1+3(1/3)+3(1/3)²+(1/3)³
= 1+1+1/3+1/27
= 2.370370...
n=4
(1+1/4)⁴
= 1+4(1/4)+6(1/4)²
+4(1/4)³+(1/4)⁴
= 1+1+3/8+1/16+1/256
= 2.44140625
n=5
(1+1/5)⁵
= 1+5(1/5)+
10(1/5)²+10(1/5)³
+5(1/5)⁴+(1/5)⁵
= 1+1+2/5+2/25+1/125+1/3,125
= 2.48832
n=6
(1+1/6)⁶
= 1+6(1/6)+
15(1/6)²+20(1/6)³+15(1/6)⁴
+6(1/6)⁵+(1/6)⁶
= 1+1+5/12+5/54+5/1296+1/1296+1/6⁶
≈ 2.5108239...
It goes to e, but slowly. The obtained expression
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
goes faster! Indeed,
n=2
sum_k 1/k! = 2.5 already
n=3
sum_k 1/k! = 2.6666...
n=4
sum_k 1/k! = 2.7083333...
n=5
sum_k 1/k! = 2.716666...
n=6
sum_k 1/k! = 2.718055555...
n=7
sum_k 1/k! ≈ 2.7182539...
This is ridiculous. It is the definition of e.
How can a limit be a number?
e isn't a number?
If there was no limit (such as growing without bounds or simply never getting closer and closer to a single value), only then would you have to worry about a limit not being a number.
@@fylthle is a constant. We need more reasoning for why we get e.
lhpotial make it way too easy
Amazingly useless.
Learn to how to write letters and numbers
Ik my handwriting is not exactly great on the board. I've been improving it lately; I apologize if it caused any trouble.
wonderful years KEVIN ARNOLD!!!!!!!
yo bruh \left(
ight) your parenthesis, that shit in the thumbnail looks diabolical
Lmao yes I had started using LaTeX recently when I recorded the video, so I didn't know there were better parenthesis. will definitely use left right parenthesis next time :)