# $w =\operatorname{arcsinh}(1+2\operatorname{arcsinh}(1+2^2\operatorname{arcsinh}(1+2^{2^2}\operatorname{arcsinh}(1+\dotsm$

In the context of positive reals consider$$\DeclareMathOperator{\arcsinh}{arcsinh}$$

$$w= \arcsinh( 1 + 2 \arcsinh( 1 + 2^2 \arcsinh ( 1 + 2^{2^2} \arcsinh( 1 + 2^{2^{2^2}} \arcsinh( 1 + \dotsm$$

Now consider a real $$A > w$$

Then the iterations

$$A_0 = A$$

$$A_1 = (\sinh(A) - 1)/2$$

$$A_2 = (\sinh(A_1) - 1)/ 2^2$$

$$...$$

grow superexponentially ! (Tetration)

Reals $$B$$ in $$[0,w[$$ do not give a sequence $$B_n$$ ( same iterations as above ) that grows to infinity.

So the interesting things are

1) the value of $$w$$

2) How fast does the sequence

$$w_0 = w$$

$$w_1 = (\sinh(w_0) - 1 ) / 2$$

$$...$$

grow ?

Could it be slower than superexponential ?

Consider the analogue

$$3 = \sqrt{1 + 2 \sqrt{ 1 + 3 \sqrt{ 1 + 4 \sqrt {...}}}}$$

Where the iterations $$c_1 = 3, c_n = ( c_{n-1}^2 - 1) / n$$ give the sequence $$3,4,5,6,7,...$$ rather than a double exponential growth ( as expected at first ).

Unfortunately tetration-like ideas tend to give Numbers too large for computation, If we are not careful. So we need a trick or some theory probably.

I tried to compute $$w$$ and arrived at the estimate

$$w = 2.613\,022\,592\,281\,8\!\dots$$

This Number might be wrong but this is my guess. Also the number seems familiar but that may be my imagination.

Is there an efficient way to compute $$w$$ ?

Are my digits correct ?

And the main question again :

How fast does $$w_n$$ grow ??

But this is in the context of reals only.

• Sorry, I modified the title after the body, and the macro to do that should have been set at the beginning of the title. It's fixed now. – Bernard Jan 31 '18 at 22:37
• I noticed Thanks – mick Jan 31 '18 at 22:42
• B.t.w., the official name of this function is asinh, or (in old style) Argsinh, or even in very old style, Argsh since sinh was denoted sh. – Bernard Jan 31 '18 at 22:47
• I have seen arcsinh used too. But Thanks for the info. In Dutch it is boogsinh :) – mick Jan 31 '18 at 22:50
• What does boog mean? – Bernard Jan 31 '18 at 22:55