In my engineering school, studying strength of materials resulted in solutions that involve trig functions. But the professor said "well, since we're engineers, we can replace all those pesky trigs with a second degree polynomial, which will render us a 3% error margin". Not for everything of course but for many, many things. That made our lives so much easier.
This isn’t necessarily an approximation. You can rigorously use Sterling’s formula from its definition. Use the mathematician’s favorite trick of “multiplying by 1,” where your version of “1” is the limit as k goes to infinity of k!/S(k) where S(k) is the approximate formula. You can then cancel out terms with no fear.
Just in case, for anyone who didn't understand why we can "approximate" here freely, it's because the difference between k! and Sterling's approximation for it goes to 0 as k goes to infinity, so they can always be interchanged inside a limit, as long as k is going to infinity, without changing the value of the limit.
[Ignore my comment, correct explantation given by the answer below] Small correction, their difference doesn't go to zero, the ratio between them goes to 1 as the input tends to infinity, it's a subtle difference but very meaningful in a lot of contexts. While the implications are the same for this example, it's more like the error in the approximation relative to what you are calculating is miniscule. For all that we know the error could be oscillating back and forth or even increasing, but not in a meaningful way in the scale of the argument.
@@healer1461 No, what I said is correct. Their difference does go to zero, which is a strictly stronger result than their ratio going to one and directly implies it. If their difference didn't go to zero, you couldn't interchange them in the limit as he did.
@@healer1461 n and n+1 also have ratio that goes to one, but because their difference doesn't go to zero, you cannot change n to n+1 in most cases without changing the limit, but if two functions have difference trending towards zero, they can always be interchange inside limit, provided you never violate functional domains.
@@newwaveinfantry8362 Now that I looked into it, I recognize you were right. I seem to have always assumed that their difference exploded because the way I'm used to approximating it, plus their graphs can be very misleading given for low order approximations.
@@newwaveinfantry8362 Just out curiosity, but does there exist a reasonable example of function F ~ H, where the limit as the input goes to infinity of F/G ≠ H/G, for some other function G? (Assuming everything converges)
I also got the joke even though I am not a engineer because I love factorials so i did some study on it btw at the start i have no idea that it would lead to stirling approximation . A W Limit fs
That approximation is used so much in soft condensed matter physics as well. I recognized it as soon as you wrote the limit 😂
In my engineering school, studying strength of materials resulted in solutions that involve trig functions. But the professor said "well, since we're engineers, we can replace all those pesky trigs with a second degree polynomial, which will render us a 3% error margin". Not for everything of course but for many, many things. That made our lives so much easier.
A small detail, it should have been floor(u)=n when n
No since it was only a one directional implication that wasn't necessary
This isn’t necessarily an approximation. You can rigorously use Sterling’s formula from its definition. Use the mathematician’s favorite trick of “multiplying by 1,” where your version of “1” is the limit as k goes to infinity of k!/S(k) where S(k) is the approximate formula. You can then cancel out terms with no fear.
Just in case, for anyone who didn't understand why we can "approximate" here freely, it's because the difference between k! and Sterling's approximation for it goes to 0 as k goes to infinity, so they can always be interchanged inside a limit, as long as k is going to infinity, without changing the value of the limit.
[Ignore my comment, correct explantation given by the answer below]
Small correction, their difference doesn't go to zero, the ratio between them goes to 1 as the input tends to infinity, it's a subtle difference but very meaningful in a lot of contexts. While the implications are the same for this example, it's more like the error in the approximation relative to what you are calculating is miniscule. For all that we know the error could be oscillating back and forth or even increasing, but not in a meaningful way in the scale of the argument.
@@healer1461 No, what I said is correct. Their difference does go to zero, which is a strictly stronger result than their ratio going to one and directly implies it. If their difference didn't go to zero, you couldn't interchange them in the limit as he did.
@@healer1461 n and n+1 also have ratio that goes to one, but because their difference doesn't go to zero, you cannot change n to n+1 in most cases without changing the limit, but if two functions have difference trending towards zero, they can always be interchange inside limit, provided you never violate functional domains.
@@newwaveinfantry8362 Now that I looked into it, I recognize you were right. I seem to have always assumed that their difference exploded because the way I'm used to approximating it, plus their graphs can be very misleading given for low order approximations.
@@newwaveinfantry8362 Just out curiosity, but does there exist a reasonable example of function F ~ H, where the limit as the input goes to infinity of F/G ≠ H/G, for some other function G? (Assuming everything converges)
This was brilliant, but would there be a way to solve it without Stirling's approximation?
I haven't found one yet but I think this is pretty good.
niceeee! That stirling approximation step was funny as well as crazy
I also got the joke even though I am not a engineer because I love factorials so i did some study on it btw at the start i have no idea that it would lead to stirling approximation . A W Limit fs
I am not an engineer but Stirling’s Formula is my favorite math result. I knew where this was going once I saw what the integral was equal to.😏
I love integrals like this.
Nice twist ending. 😎
This was beyond crazy 🔥🔥💯💯
Nice! Now you add one more for your to do list,stirling approximation proof 😂
I heard this approximation is correct when k goes to infinity, right?
It means the limit of (Stirling's approx)/k! as k tends to infinity is 1. That's what we mean by asymptotically equal.
@@maths_505thanks😊
Near enough is good enough.
- The First Fundamental Theorem of Engineering
Ah yes... my favorite method of solving problems, hacks.
hey that approximation doesnt work on wolfram alpha - as k approaches infinity , that value also seems to go to infinity - what could have gone wrong?
The approximation working means that the limit of (Stirling's approx)/(k!) as k tends to infinity is 1.
It's indeed crazy!
Thank you for this video
Sometimes life brings nice surprise... Merry Christmas!
On behalf of all my fellow engineers, thank you for that. 🙂
That was truly awesome
Summation is all about telescoping
Engineers approximate pi to 3 .... 😅, but some how the World still there 🙃
approximations are art