Calculating infinite series of roots

[KeithB]

https://skullsinthestars.com/2019/08/18/...-identity/

Is there a way to calculate this easily on the Prime?

i.e. showing that (2^.5)^(2^.5)^(2^.5)... = 2?

[Helge Gabert]

There is no symbol for repeated exponentiation on the calc, as opposed to the repeated summation and product operator, so I'm not sure if you can show that symbolically with the limit function.

But, of course, you can always write a program and show that numerically, you get very close to 2. For example, with a REPEAT ... UNTIL loop, using approx(sqrt(2)), and setting your tolerance at 1E-12, you get to 2 after 71 iterations, after which the result doesn't change any more.

[ijabbott]

A "power tower" of \(n\) \(x\)'s \(\underbrace{x^{x^{x^{\dots^x}}}}_n\) is also known as the \(n\)th tetration of \(x\). The operation is called "tetration" because it is the fourth in a series of operations: addition, multiplication, exponentiation, tetration. There are other terms for the same thing. There are also several notations, but one such notation is \(^{n}x\).

Now we come on to the "infinite tetration" of \(x\), or \(^{\infty}x\), which is \(\lim\limits_{n\to\infty}{^{n}x}\). For certain real values of \(x\) such as \(x=\sqrt{2}\), this infinite tetration converges (to \(2\) in this case). In fact it only converges for real \(x\) in the interval \(\big[e^{-e}, e^\frac{1}{e}\big]\) (approx. \([0.066, 1.445]\)).

The limit of convergence can evaluated using the Lambert W function:

\[^{\infty}x = \lim_{n\to\infty}{^{n}x} = \frac{W(-\ln(x))}{-\ln(x)} \big|_{e^{-e} \le x \le e^\frac{1}{e}}\]

where the Lambert W function is defined by \(W(z{e^z})=z\), or by \(z_0 = W(z_0)e^{W(z_0)}\).

Unfortunately, the Lambert W function is not yet built-in on the HP Prime, although it is available in Xcas/Giac, so maybe later....

[Helge Gabert]

(08-30-2019 11:57 PM)ijabbott Wrote:  In fact it only converges for real \(x\) in the interval \(\big[e^{-e}, e^\frac{1}{e}\big]\) (approx. \([0.066, 1.445]\)).

Yes, very true, and that was shown by Euler (back in the days . . . ).

Support for the Lambert W function would be nice, if there is ever an update again!

[Stevetuc]

(08-31-2019 12:59 AM)Helge Gabert Wrote:  

(08-30-2019 11:57 PM)ijabbott Wrote:  In fact it only converges for real \(x\) in the interval \(\big[e^{-e}, e^\frac{1}{e}\big]\) (approx. \([0.066, 1.445]\)).

Yes, very true, and that was shown by Euler (back in the days . . . ).

Support for the LambertW function would be nice, if there is ever an update again!

This cas program uses fsolve to calc Lambert fn

Code:


#cas
lmb(z):=
fsolve(equal(w*e^w-z,0),w)
#end

Graph from wikipedia


If z <0 lmb(z) returns both principal and negative branch solutions :

lmb(-0.5/e) returns [−2.67834699002,−0.231960952987]

lmb(-1/e) returns −0.999999842597 (negative and principal branch converge at -1 when z=-1/e)

lmb(1) returns 0.56714329041

lmb(2) returns 0.852605502014

[ijabbott]

(08-30-2019 11:57 PM)ijabbott Wrote:  The limit of convergence can evaluated using the Lambert W function:

\[^{\infty}x = \lim_{n\to\infty}{^{n}x} = \frac{W(-\ln(x))}{-\ln(x)} \big|_{e^{-e} \le x \le e^\frac{1}{e}}\]

where the Lambert W function is defined by \(W(z{e^z})=z\), or by \(z_0 = W(z_0)e^{W(z_0)}\).

I've just noticed that this evaluates to \(\frac{0}{0}\) for \(x=1\). Hmm... more work needed? Also when \(1 \lt x \le e^\frac{1}{e}\), then letting \(z=-\ln(x)\), \(-\frac{1}{e} \le z \lt 0\), and there are two branches of the \(W\) function in this interval. I guess it uses the upper branch?