I have induced convergence!
We use Ackermann numbers!
\( \vartheta_n(s) = \frac{\sin( \pi (s- n))}{ \pi(s-n)}(n \,\,\bigtriangleup_n\,\,n)^{s-n} \)
This is a much more sophisticated function.
These grow faster than anything in the planet. Certainly faster than \( x \,\,\bigtriangleup_n\,\,y \). Unfortunately; no clue how to prove this. I know that it decreases fast enough to zero to dampen the growth of hyper operators. Because it grows faster than any hyper operator. Sort of how n! grows faster than any natural iterated multiplication of x; \( x^n \).
\( x\,\,\bigtriangleup_s\,\,y = \prod_{n=0}^{\infty} (x\,\,\bigtriangleup_n\,\,y)^{\vartheta_n(s)} \)
Now to tackle recursion. I will need a third Aha! to conquer this one. Anyone have any tips on how to prove this converges? I think the ratio test on this one is pretty useless.
We use Ackermann numbers!
\( \vartheta_n(s) = \frac{\sin( \pi (s- n))}{ \pi(s-n)}(n \,\,\bigtriangleup_n\,\,n)^{s-n} \)
This is a much more sophisticated function.
These grow faster than anything in the planet. Certainly faster than \( x \,\,\bigtriangleup_n\,\,y \). Unfortunately; no clue how to prove this. I know that it decreases fast enough to zero to dampen the growth of hyper operators. Because it grows faster than any hyper operator. Sort of how n! grows faster than any natural iterated multiplication of x; \( x^n \).
\( x\,\,\bigtriangleup_s\,\,y = \prod_{n=0}^{\infty} (x\,\,\bigtriangleup_n\,\,y)^{\vartheta_n(s)} \)
Now to tackle recursion. I will need a third Aha! to conquer this one. Anyone have any tips on how to prove this converges? I think the ratio test on this one is pretty useless.
