a nice result from Ramanujan's "lost notebook"
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- เผยแพร่เมื่อ 11 ก.พ. 2025
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How genius Ramanujam was? He thought of hundreds of hard problems like this that nobody asked before and solved them just for the sake of it.
He wasn't t a genius he said god whispered him these ideas ramanujan just implemented them
@@mrhatman675 * *Ramanujan* His name is capitalized, just as your actual name is.
@@forcelifeforce who cares
Both a genius and god spoke to him
@@SuperSilver316 No god told him the ideas ramanujan just implemented them
so basically Ramanujan's approach was "here's a complicated expression, I'll just make a way more complicated version of the same, find a way to connect the more complicated versions in a chain of relationships, and the fact that this chain is infinite somehow lets us push the last infinitely complicated term out of relevance. And bonus points for actually converging sometimes"
In terms of ancient memes, Yo dawg I heard you liked infinity, so we put infinity inside your infinity, so you can formal series while you formal series
That was actually such a trip!
I don't know much about power series where the exponents have degree higher than 2, but one of the main reasons that sums like Σx^n^2 have been studied so much is due to their connection with modular forms (in particular, theta functions). The most basic theta function identities come from the Gaussian integral ∫e^(-x^2)dx, and that's where the square exponents come from.
Looks another result that we can use to show that Grandi’s series has an Abel Sum of 1/2.
20:46
Each individual step is quite simple but the amount of work it must have taken to figure out this particular order of steps and clarify them must've been insane. Just endless playing with slightly different forms.
The way you put it here is pretty structured. I bet originally it very much was not.
Bro really knew infinity
Missing a step to show that L_n tends to zero for n tending to infinity, in the radius for mod x less than 1.
it's left as a trivial exercise for the reader
The margins of the notebook were too small
Ramanujan in a nutshell
Typo' in original left-hand side, third numerator: "(1-x^2)" should be "(1-x^3)".
My numerical analysis can confirm you. I previously had a deviation between the convergence values. - Thanks for the correction.
Me parece maravilloso tu contenido estimado, saludos desde chile
19:17 forgot to write ( - x^8)
Wow
wow!
Obscure
I cant see the pattern in the question though
cant understand but cant look away either^^
I like the chalk colorscheme on this channel. Looks like xoria256.
Gee, there's a much simpler way to show this....nah, just kidding!
I guess "hypergeometric series" is a misnomer here, because it is generally referred to another thing (en.wikipedia.org/wiki/Hypergeometric_function#The_hypergeometric_series). The series in the RHS of the identity actually resembles that of a version of theta function
I believe he meant 'basic hypergeometric series' which are essentially q-analogs of hypergeometric series.
Ramanujam style! No sympy, no mathcad, no mapple, ...
Wonder if he at least had multi-colored chalk?
Please write in English words.
@@forcelifeforcethey are. Those are maths programs, tho I assume they've spelt maple incorrectly