Excellent video, you've made your reasoning very clear and easy to follow That said, here's my comments about the amateur method. My suggestions might actually be from the professional method but I'm not sure It's not that there's anything wrong with the amateur method, it's just that it feels incomplete Sure, the square root of a number will never be zero unless its input is zero, but that's the case for a lot of functions. And the case of infinitely many of them adds the wrinkle of limits and convergence Like x^2 can only be 0 if x = 0, but if I start with 0.5 and continue squaring forever I end up at 0 anyway It turns out with the function g(y) = sqrt(xy), y will converge to y = x and not y = 0
I am a little bit hesitant about the correctness of the solution, while it feels like a nice argument and in fact the result is correct, there are few objections that I want to make to the proof: 1) Infinite expressions like sqrt(x*sqrt(x*sqrt(x*sqrt(x*...)))) or x^x^x^x^... are tricky to play with substitution, because it is not known from before whether they actually converge to a number or not. They might not converge to any real number and hence assuming we can play with them as real numbers can lead to a lot of different paradoxes. (Watch this to understand better why this is true: ua-cam.com/video/DmP3sFIZ0XE/v-deo.html ) 2) Lets suppose that we know the expression inside our integral converges to a real number for each x, still we wouldn't be able to discard the case of y = 0 because even though it is trivial to see that y = 0 when x = 0, it is not easy to see whether any other real value x does not make the infinite expression to converge to 0. A better way to deal with this integral is by making the expression: sqrt(x*sqrt(x*sqrt(x*...))) = x^(1/2) * x ^ (1/4) * x ^ (1/8) * ... = x ^ (1/2 + 1/4 + 1/8 + ...) = x ^ (1) = x => integral(x) = (1/2)*x^2 + C
A final trick would be to rewrite the answer as something with an infinite nested radical. I can think of a couple of ways to do that, but I'm not sure which one is "right", if any. (x/2) y replacing y with that infinite radical is one way. (y)(y)/2 is another - two nested radicals for the price of one! Not sure might happen with an infinite number of applications of the chain rule to the alternate form in this video's follow-up. That just makes my head hurt.
Beautifully explained. Keep up the good work and # of subscribers will explode!
Playing with infinity is so much fun! Cool proof.
Excellent video, you've made your reasoning very clear and easy to follow
That said, here's my comments about the amateur method. My suggestions might actually be from the professional method but I'm not sure
It's not that there's anything wrong with the amateur method, it's just that it feels incomplete
Sure, the square root of a number will never be zero unless its input is zero, but that's the case for a lot of functions. And the case of infinitely many of them adds the wrinkle of limits and convergence
Like x^2 can only be 0 if x = 0, but if I start with 0.5 and continue squaring forever I end up at 0 anyway
It turns out with the function g(y) = sqrt(xy), y will converge to y = x and not y = 0
I am a little bit hesitant about the correctness of the solution, while it feels like a nice argument and in fact the result is correct, there are few objections that I want to make to the proof:
1) Infinite expressions like sqrt(x*sqrt(x*sqrt(x*sqrt(x*...)))) or x^x^x^x^... are tricky to play with substitution, because it is not known from before whether they actually converge to a number or not. They might not converge to any real number and hence assuming we can play with them as real numbers can lead to a lot of different paradoxes. (Watch this to understand better why this is true: ua-cam.com/video/DmP3sFIZ0XE/v-deo.html )
2) Lets suppose that we know the expression inside our integral converges to a real number for each x, still we wouldn't be able to discard the case of y = 0 because even though it is trivial to see that y = 0 when x = 0, it is not easy to see whether any other real value x does not make the infinite expression to converge to 0.
A better way to deal with this integral is by making the expression: sqrt(x*sqrt(x*sqrt(x*...))) = x^(1/2) * x ^ (1/4) * x ^ (1/8) * ... = x ^ (1/2 + 1/4 + 1/8 + ...) = x ^ (1) = x => integral(x) = (1/2)*x^2 + C
Solid observations. Thank you for your contribution.
no lies you are one of the best.
A final trick would be to rewrite the answer as something with an infinite nested radical. I can think of a couple of ways to do that, but I'm not sure which one is "right", if any. (x/2) y replacing y with that infinite radical is one way. (y)(y)/2 is another - two nested radicals for the price of one!
Not sure might happen with an infinite number of applications of the chain rule to the alternate form in this video's follow-up. That just makes my head hurt.
Thanks Sir
Wow 😮
Awesome!
Can I get the professional method sir
We can use laws of exponents and some Infinite gp
Sir I emailed you a list of questions please do check them out, just ignore the one with the quadratic equation, because I discovered an error in it
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