# Puzzling math problem

**URL:** <https://boards.straightdope.com/t/puzzling-math-problem/207332>\
**Category:** Factual Questions\
**Created:** [October 14, 2003, 6:41am UTC](https://boards.straightdope.com/t/puzzling-math-problem/207332 "2003-10-14T06:41:37Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![agiantdwarf](https://avatars.discourse-cdn.com/v4/letter/a/8c91f0/32.png) [@agiantdwarf](https://boards.straightdope.com/u/agiantdwarf)\
**Post date:** [October 14, 2003, 6:41am UTC](https://boards.straightdope.com/t/puzzling-math-problem/207332/1 "2003-10-14T06:41:37Z")

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(This isn’t my homework.)

Evaluate the limit of this: [http://users.adelphia.net/~ybechor/problem.GIF](http://users.adelphia.net/~ybechor/problem.GIF)

The solution is:

R = root(2)^(root(2)^(root(2)…  
R = root(2)^R

Therefore, R= 2.

Here’s the puzzling part: R can also equal 4. How can this be? A number should only have one limit.

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**Author:** ![MartinL](https://avatars.discourse-cdn.com/v4/letter/m/ebca7d/32.png) [@MartinL](https://boards.straightdope.com/u/MartinL)\
**Post date:** [October 14, 2003, 7:10am UTC](https://boards.straightdope.com/t/puzzling-math-problem/207332/2 "2003-10-14T07:10:00Z")

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Let’s say what you have is a _sequence_, and it is defined _recursively_. We can apply _Banach_’s theorem about fixed points.  
starting point: a(1) = root(2)  
recursion: a(i) = root(2) ^ a(i-1)  
In order to apply the theorem, we will show the following:  
a = 2 is a fixed point (obviously: root(2) ^ 2 = 2)  
a(i) will grow towards 2: a(i) \> a(i-1) (true, because when we raise x to the power of y, and x\>1, y\>1 then x^y\>x)  
a(i) will not grow beyond 2: a(i) \< 2 (true, because the power needed to raise root(2) beyond 2 is greater than 2, so once smaller than 2, forever smaller than 2)  
That’s it! We proved that the limit of your sequence is 2.

Oh yes: The value 4 is also a fixed point for this recursive definition: root(2) ^ 4 = 4, but you will need a different starting point. With the information given, you may be able to consider different ranges of starting points and their limits.

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**Author:** ![MartinL](https://avatars.discourse-cdn.com/v4/letter/m/ebca7d/32.png) [@MartinL](https://boards.straightdope.com/u/MartinL)\
**Post date:** [October 14, 2003, 7:15am UTC](https://boards.straightdope.com/t/puzzling-math-problem/207332/3 "2003-10-14T07:15:08Z")

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Sorry, I forgot:

> [@](#):
>
> \*Originally posted by agiantdwarf \*  
> \*\*Here’s the puzzling part: R can also equal 4. How can this be? A number should only have one limit. \*\*

A _number_ has no limit, a number is simply a number. A _sequence_ or a _series_ can have one limit at most (in real numbers). An _equation_ like R = root(2) ^ R on the other hand can have any number of _solutions_.

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**Author:** ![MartinL](https://avatars.discourse-cdn.com/v4/letter/m/ebca7d/32.png) [@MartinL](https://boards.straightdope.com/u/MartinL)\
**Post date:** [October 14, 2003, 7:20am UTC](https://boards.straightdope.com/t/puzzling-math-problem/207332/4 "2003-10-14T07:20:17Z")

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Another follow-up:

> [@](#):
>
> \*Originally posted by agiantdwarf \*  
> \*\*R = root(2)^(root(2)^(root(2)…  
> R = root(2)^R  
> \*\*

The first line defines R as the limit of a sequence. The second line defines R as (one of the) solution(s) of an equation. The two lines are related: Any solution from the second line is a good candidate as a solution of the first line, but that’s it. The lines are not equivalent.