# Tetration Forum

Full Version: What am I doing wrong here?
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I was looking at tetration and considering its little taylor series, given by:
(1) $sexp(x) = \bigtriangleup \sum_{n=0}^{\infty} a_n + n(x \bigtriangledown l) - ln(n!)$

where:
$\bigtriangleup \sum_{n=0}^{r} f(n)= f(0) \bigtriangleup f(1) \bigtriangleup...{f®}$
and:
$x \bigtriangleup y= \ln(e^x + e^y)$
$x \bigtriangledown y = \ln(e^x - e^y)$

specifically, what I was trying to do is find a relationship between the coefficients of tetrations' little taylor series and normal taylor series. We'll say:
$sexp(x) = \sum_{n=0}^{\infty} b_n \frac{(x-l)^n}{n!}$

so I'm looking for $b_n$ as an expression of $a_n$ or vice versa.

So I started off by looking at (1):

$sexp(x) = \bigtriangleup \sum_{n=0}^{\infty} a_n + n(x \bigtriangledown l) - ln(n!)$

$sexp(x) = ln(\sum_{n=0}^{\infty} e^{a_n + n(x \bigtriangledown l) - ln(n!)})$

$sexp(x+1) = \sum_{n=0}^{\infty} e^{a_n} \frac{(e^x - e^l)^n}{n!}$

and now let this equal our formula for sexp(x+1) using $b_n$ or the normal taylor series.

$\sum_{n=0}^{\infty} e^{a_n} \frac{(e^x - e^l)^n}{n!} = \sum_{d=0}^{\infty} b_d \frac{(x+1-l)^d}{d!}$

and subtract the right hand side from the left hand side

$\sum_{n=0}^{\infty} \frac{e^{a_n}(e^x - e^l)^n - b_n (x+1-l)^n}{n!} = 0$

and now, since x is essentially arbitrary, let x = l to give:
$\sum_{n=0}^{\infty} \frac{b_n}{n!} = 0$

but this contradicts the original Taylor series expansion
$sexp(x) = \sum_{n=0}^{\infty} b_n \frac{(x-l)^n}{n!}$

which states:
$\sum_{n=0}^{\infty} \frac{b_n}{n!} = sexp(l+1)$

Any help would be greatly appreciated, thanks.

The only solution I have is $l = -2$
(05/12/2011, 07:53 PM)JmsNxn Wrote: [ -> ]$x \bigtriangledown y = \ln(e^x - e^y)$

One problem could be that if you let x=l then $x \bigtriangledown l = -\infty$. And calculating with infinities is always somewhat risky.
oh, that's obvious there in the first equation. I hate missing stuff that's right in front of me.

yeah, I've really started to notice that infinity is difficult to work with. It frustrated me at first but then I realize it comes with the territory.