To everyone who's concerned about the circular reasoning in this video-yes, the proof involves some circular reasoning. As I already told someone in the comment section, here's my take on this: there are two important facts here: 1) the derivative of lnx is one over x, and 2) the limit I showed in this video equals e. After some research, I believe it is impossible to prove one fact without claiming that the other one is true IF you're only using limits to prove these facts. In proving that the derivative of lnx is one over x, every proof I've seen out there uses the fact that the limit shown in this video equals e. Conversely, every proof showing that the video's limit equals e either uses L'Hopital's Rule or the method I presented here, both of which rely on the derivative of lnx being equal to one over x. So yes, this indeed feels like circular reasoning, but I didn't realize this when I recorded the video, so thanks to everyone who's highlighting it! There may be a proof out there of one of these two facts that doesn't use a limit approach. This may result in a proof of one of the two facts that doesn't rely on assuming the other is true, which would be quite interesting to see.
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.
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.
While others are criticizing you, ill appreciate you for your efforts. Keep going you'll do well. Also only take constructive criticism. The ones who are just making fun of you are jerks. Dont be a d³x/dt³.
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
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.
In advanced texts you start by defining ln x=\int_1^x dt/t, and e^x as the inverse function. Then everything follows much easier than in the standard approach. Of course, eventually one shows that exp(x)=e^x where e=exp(1). This particular presentation is childish.
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?
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
@@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.
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
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.
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...
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.
??? you use the definition of e to proof that definition?;)))) bernoulli defined e is the limit of that function when he had the problem of continuous compounding of interest
acob Bernoulli discovered this constant in 1683, while studying a question about compound interest:[5] An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year? If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.44140625, and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035.... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00 × (1 + 1/n)n.[20][21] Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals.[5] Compounding weekly (n = 52) yields $2.692596..., while compounding daily (n = 365) yields $2.714567... (approximately two cents more). The limit as n grows large is the number that came to be known as e. That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. Here, R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05.[20][21] cre:wikipedia
Her equation ki ek limit hoti he bad me vo galat sabhit hote he kyoiki ye sirf ek mathematic hi he or kuch nahi hota he 😅equation se duniya chalati nahi he 😅vastvikata se sambandh rakhana chahiye kalpana se nahi 😅partikal ka astitva hota he 😅partikal ek hakat he 😅dimension ka nahi 😅partikal ko dimension he dimension se partikal nahi 😅partikal se duniya chalati he 😅dimension se nahi 😅partikal ka bhavatik rup se astitva hota he 😅baki Sam hamare man ka bram 😅
Sorry, but this video is totally wrong.The problem is that to find out what the derivative of a logaritm is, you have to know the value of the limit itself. So, to prove your limit, you cannot use any result that depends on the knowing of the limit itself, this means you canno use De l'Hopital. This is called circular reference and cannot be used as a proof. Please delete this video as fast as you can.
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 :)
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
To everyone who's concerned about the circular reasoning in this video-yes, the proof involves some circular reasoning. As I already told someone in the comment section, here's my take on this: there are two important facts here: 1) the derivative of lnx is one over x, and 2) the limit I showed in this video equals e. After some research, I believe it is impossible to prove one fact without claiming that the other one is true IF you're only using limits to prove these facts. In proving that the derivative of lnx is one over x, every proof I've seen out there uses the fact that the limit shown in this video equals e. Conversely, every proof showing that the video's limit equals e either uses L'Hopital's Rule or the method I presented here, both of which rely on the derivative of lnx being equal to one over x. So yes, this indeed feels like circular reasoning, but I didn't realize this when I recorded the video, so thanks to everyone who's highlighting it! There may be a proof out there of one of these two facts that doesn't use a limit approach. This may result in a proof of one of the two facts that doesn't rely on assuming the other is true, which would be quite interesting to see.
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.
While others are criticizing you, ill appreciate you for your efforts. Keep going you'll do well. Also only take constructive criticism. The ones who are just making fun of you are jerks. Dont be a d³x/dt³.
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
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
In advanced texts you start by defining ln x=\int_1^x dt/t, and e^x as the inverse function. Then everything follows much easier than in the standard approach. Of course, eventually one shows that exp(x)=e^x where e=exp(1). This particular presentation is childish.
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?
Teaching many concepts in 1 video. Thank you very much!
e = n^n / (n-1)^n for n --> infinity
This is NOT just a way to define e. This is the ORIGINAL way.
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.
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
Really liked your explanation✌️please do more videos on how natural logarithmic table was first made
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.
Your explanation is dammm good....teach me also in some video😅😅plz plz
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...
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.
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.
??? you use the definition of e to proof that definition?;)))) bernoulli defined e is the limit of that function when he had the problem of continuous compounding of interest
acob Bernoulli discovered this constant in 1683, while studying a question about compound interest:[5]
An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?
If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.44140625, and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035.... If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00 × (1 + 1/n)n.[20][21]
Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals.[5] Compounding weekly (n = 52) yields $2.692596..., while compounding daily (n = 365) yields $2.714567... (approximately two cents more). The limit as n grows large is the number that came to be known as e. That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of R will, after t years, yield eRt dollars with continuous compounding. Here, R is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, R = 5/100 = 0.05.[20][21] cre:wikipedia
Her equation ki ek limit hoti he bad me vo galat sabhit hote he kyoiki ye sirf ek mathematic hi he or kuch nahi hota he 😅equation se duniya chalati nahi he 😅vastvikata se sambandh rakhana chahiye kalpana se nahi 😅partikal ka astitva hota he 😅partikal ek hakat he 😅dimension ka nahi 😅partikal ko dimension he dimension se partikal nahi 😅partikal se duniya chalati he 😅dimension se nahi 😅partikal ka bhavatik rup se astitva hota he 😅baki Sam hamare man ka bram 😅
We'll information good show and 😅😅
lhpotial make it way too easy
Amazingly useless.
👍
Sorry, but this video is totally wrong.The problem is that to find out what the derivative of a logaritm is, you have to know the value of the limit itself. So, to prove your limit, you cannot use any result that depends on the knowing of the limit itself, this means you canno use De l'Hopital. This is called circular reference and cannot be used as a proof. Please delete this video as fast as you can.
This is ridiculous. It is the definition of e.
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.
The boy is too impetuous. I don't think he explains anything. He's just showing off.
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 :)