# Tetration function in Wolfram Alpha?

**URL:** <https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962>\
**Category:** Questions\
**Tags:** wolfram-alpha\
**Created:** [July 17, 2017, 6:27pm UTC](https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962 "2017-07-17T18:27:36Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![alexanderwolters](https://avatars.discourse-cdn.com/v4/letter/a/b19c9b/32.png) [@alexanderwolters](https://community.wolfram.com/u/alexanderwolters)\
**Post date:** [July 17, 2017, 6:27pm UTC](https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962/1 "2017-07-17T18:27:36Z")

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Is there any way to enter tetration in WolframAlpha, in such a way that I can use it with the limit function? More specific, can I make Wolfram Alpha calculate the limit of sqrt(2)^sqrt(2)^sqrt(2)^sqrt(2)^… You get it? And i know it will be 2, i just wanted to ask for the general case.

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**Author:** ![sukrause](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sukrause/32/41348_2.png) [@sukrause](https://community.wolfram.com/u/sukrause)\
**Post date:** [July 20, 2017, 4:33pm UTC](https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962/2 "2017-07-20T16:33:59Z")

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This can be seen as a `FixedPoint[]`, after looking into what Mathematica makes out of it

```
In[27]:= Sqrt[2]^Sqrt[2]^Sqrt[2]^Sqrt[2]^Sqrt[2]^Sqrt[2] // FullForm
Out[27]=Power[2,Power[2,Plus[-1,Power[2,Plus[-1,Power[2,Plus[-1,Power[2,Plus[-1,Power[2,Rational[-1,2]]]]]]]]]]]

```

and keeping in mind that

```
In[18]:= ((((Sqrt[2]^Sqrt[2])^Sqrt[2])^Sqrt[2])^Sqrt[2])^Sqrt[2] // N
Out[18]= 7.10299

```

diverges, one finds the expression to reach the fixed point is

```
In[30]:= 2^FixedPoint[Power[2, Plus[-1, #]] &, Sqrt[2.]]
Out[30]= 2.

```

which can - to return to the question - be entered into W|A. Note

```
In[31]:= FixedPoint[Power[2, Plus[-1, #]] &, Sqrt[2.]]
Out[31]= 1.

```

to no surprize.

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**Author:** ![aandresg31](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/aandresg31/32/25786_2.png) [@aandresg31](https://community.wolfram.com/u/aandresg31)\
**Post date:** [July 20, 2017, 10:47pm UTC](https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962/3 "2017-07-20T22:47:24Z")

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Slightly off-question but I think this can help. I actually [did some research on tetration (and extending it to complex realms)](https://wolfram.discourse.group/t/10861/1) some weeks ago and I learnt the following through this [Wolfram MathWorld page](http://mathworld.wolfram.com/PowerTower.html).

The value of the infinite power tower  
 ![Inline34.gif](https://global.discourse-cdn.com/wolfram/original/2X/e/e5f323434fc4b6a53a4ed7a82847775658831f53.gif)  
can be computed analytically by writing

![NumberedEquation9.gif](https://global.discourse-cdn.com/wolfram/original/2X/9/9cb5a5e1deac1c903e2de595f3df3a93af6e9e45.gif)

Taking the logarithm of both sides and plugging back in to obtain

![NumberedEquation10.gif](https://global.discourse-cdn.com/wolfram/original/2X/1/1b5c8787388414519bcb5f48ae4194ac485082a4.gif)

Solving for h(z) gives

![NumberedEquation11.gif](https://global.discourse-cdn.com/wolfram/original/2X/7/768e66a6f241dc7a1a7240a592caf77a6c3cb61f.gif)  
.

Where **W(z)** is the Lambert W-function (Corless et al. 1996). h(z) converges iff e^(-e)\<=x\<=e^(1/e) (0.0659\<=x\<=1.4446; OEIS A073230 and A073229), as shown by Euler (1783) and Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35). This function is already in mathematica with the following name:

```
    ProductLog[x]

```

Just to prove it works:

 ![sdsd.PNG](https://global.discourse-cdn.com/wolfram/original/2X/2/2f94729b7f163df45636fd51bbdf96591a919b98.png)

I don’t know what things you’re working on but probably this method will be more efficient than solving for the limit manually. Hope it helped 🙂

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<div class="post-metadata">

**Author:** ![sukrause](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sukrause/32/41348_2.png) [@sukrause](https://community.wolfram.com/u/sukrause)\
**Post date:** [July 22, 2017, 10:24am UTC](https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962/4 "2017-07-22T10:24:19Z")

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Observe - despite this argument has a weakness if repeated - that (because of \infty-1=\infty) one might single out one term \sqrt2 from `a=Sqrt[2]^Sqrt[2]^Sqrt[2]^...` coming to

```
In[1]:= Solve[Sqrt[2]^a == a, a]
Out[1]= {{a -> 2}, {a -> -((2 ProductLog[-1, -(Log[2]/2)])/Log[2])}}

```

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<div class="post-metadata">

**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\
**Post date:** [July 23, 2017, 3:49pm UTC](https://community.wolfram.com/t/tetration-function-in-wolfram-alpha/10962/5 "2017-07-23T15:49:59Z")

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One can enter `solve sqrt(2)^x=x` and obtain a [result from W|A](http://www.wolframalpha.com/input/?i=solve%20sqrt%282%29%5Ex=x). It seems that W|A will not handle the more general case of a^x=x, though Mathematica is fine with it.

```
In[63]:= Solve[a^x == x, x]

During evaluation of In[63]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out[63]= {{x -> -(ProductLog[-Log[a]]/Log[a])}}

```

This does not fully address the question since it is a multivalued result, so one must figure out which case is correct for the limit problem at hand.