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 Tetration extension for bases between 1 and eta bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 11/08/2009, 02:44 PM (This post was last modified: 11/08/2009, 02:47 PM by bo198214.) (11/07/2009, 11:30 PM)dantheman163 Wrote: If we set $f(x+1)=b^{f(x)}$ which is to say $\lim_{k\to \infty} (log_{b}^{ok}((x+1)({}^k b- {}^{(k-1)} b)+{}^k b) ) = \lim_{k\to \infty} (log_{b}^{o(k-1)}(x({}^k b- {}^{(k-1)} b)+{}^k b) )$ then reduce it to $\lim_{k\to \infty} (log_{b}((x+1)({}^k b- {}^{(k-1)} b)+{}^k b) ) = \lim_{k\to \infty} (x({}^k b- {}^{(k-1)} b)+{}^k b)$ Then just strait up plug in infinity for k we get $log_{b} {}^\infty b = {}^\infty b$ which is the same as ${}^\infty b = {}^\infty b$ Slowly, slowly. The first line is what you want to show. You show from the first line something true, but you can also show something true starting from something wrong; so thats not sufficient. Also it seems as if you confuse limit equality with sequence equality. Lets have a look at the inverse function $g=f^{-1}$, $g(x)=\lim_{k\to\infty} \frac{\exp_b^{\circ k}(x)-{^k b}}{{^k b}-({^{k-1} b})}$ it should satisfy $g(b^x)=g(x)+1$. Then lets compute $g(b^x)-g(x)= \lim_{k\to\infty}\frac{\exp_b^{\circ k+1}(x)-{^k b}}{{^k b}-({^{k-1} b})} - \frac{\exp_b^{\circ k}(x)-{^k b}}{{^k b}-({^{k-1} b})} =\lim_{k\to\infty} \frac{\exp_b^{\circ k+1}(x) - \exp_b^{\circ k}(x)}{{^k b}-({^{k-1} b})}$ Take for example $x=1$ then the right side converges to the derivative of $b^x$ at the fixed point; and not to 1 as it should be. This is the reason why the formula is only valid for functions that have derivative 1 at the fixed point, e.g. $e^{x/e}$, i.e. $b=e^{1/e}$. Quote:This is really weird because if i do $\lim_{k\to \infty} (log_{b}^{ok}(1({}^k b- {}^{(k-1)} b)+{}^k b) )$ for $b= sqrt2$ i get about 1.558 which is substantially larger then $sqrt2$ Can anyone else confirm that $\lim_{k\to \infty} (log_{b}^{ok}(1({}^k b- {}^{(k-1)} b)+{}^k b) ) \approx 1.558$ for $b= sqrt2$ ? Try the same with $b=e^{1/e}$ and it will work; but for no other base; except you use the modified formula I described before. « Next Oldest | Next Newest »

 Messages In This Thread Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 03:00 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/05/2009, 01:44 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 11:53 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 09:31 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 05:11 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 08:12 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/07/2009, 11:30 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/08/2009, 02:44 PM RE: Tetration extension for bases between 1 and eta - by mike3 - 11/12/2009, 07:11 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:01 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/15/2009, 01:40 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:48 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/17/2009, 02:40 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/17/2009, 10:59 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/19/2009, 05:06 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/19/2009, 10:55 AM

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