What type of sequence is this; can you express it using a sum or product?

[Saracen Rue]

TL;DR Summary

What type of sequence is ##(\frac{1}{5}+(\frac{1}{5}+(\frac{1}{5}+(...)^2 )^2)^2)^2##?

Hi all; I have a very basic understanding of sequences and series and recently encountered a sequence which really has me confused: $$(\frac{1}{5}+(\frac{1}{5}+(\frac{1}{5}+(...)^2 )^2)^2)^2$$ What type of sequence would you call this? I couldn't even google it because I couldn't work out how to describe what kind of pattern this is.

I'm hoping I can find out more about what kind of sequence this is and if it's possible to define it as being the sum or product of something. Any help is greatly appreciated.

 

[Orodruin]

Assuming the recursion is infinite, it should be clear that it is of the form
$$
x = (1/5 + x)^2.
$$
Solve for ##x##.

 

[member 587159]

I couldn't find the proper english term, but I would call it "nested expressions". Compare with https://en.wikipedia.org/wiki/Continued_fraction

 

[member 587159]

Assuming the recursion is infinite, it should be clear that it is of the form
$$
x = (1/5 + x)^2.
$$
Solve for ##x##.

It should be noted that convergence must be checked here. This just shows that IF there is a limit, then it must satisfy this equation. Moroever, such an expression can give you at most one solution and this quadratic will give you two, so certainly one of them will not qualify.

 

[Saracen Rue]

Thank you both a lot, you've been very helpful :)

 

[Orodruin]

It should be noted that convergence must be checked here. This just shows that IF there is a limit, then it must satisfy this equation.

I would say this depends on how you interpret the expression. If you interpret it as a recursive sequence such as
$$
x_{n+1} = (1/5 + x_n)^2,
$$
then this is certainly true. However, in general, sequences on the form ##x_{n+1} = f(x_n)## can have several different limits depending on how ##x_0## is chosen. Either of the solutions to ##x = f(x)## will be a fixed point, but generally may or may not be a limit when starting at nearby points. There is also the clear possibility of the sequence diverging towards positive infinity as ##x_{n+1} \sim x_n^2## for large ##|x_n|##.

An alternative interpretation is to just interpret the expression as representing "a number for which ##x = (1/5 + x)^2##". In that case, any of the two solutions will do, but you may or may not find the number by using the recursion above.

 

[member 587159]

I implicitely assumed that it was the limit of a recursively defined sequence, but I should have been more clear!

An alternative interpretation is to just interpret the expression as representing "a number for which ##x = (1/5 + x)^2##". In that case, any of the two solutions will do, but you may or may not find the number by using the recursion above.

This I have never seen before. In some cases (not here) such a quadratic can even give a negative number and allowing a negative number as result of this seems odd and not very useful.

 

[Stephen Tashi]

Summary:: What type of sequence is ##(\frac{1}{5}+(\frac{1}{5}+(\frac{1}{5}+(...)^2 )^2)^2)^2##?

In general, notation indicating a infinitely nested expression is not a sequence. Such notation doesn't necessarily denote any mathematical object. You are correct that the usual way to define the meaning of an infinitely nested expression is to define it to be the limit of a certain sequence. However, it may be possible to interpret an infinitely nested expression as a sequence in more than one way.

A good video on this topic is:

An interesting point covered in that video is that the notation for infinite continued fractions has two distinct plausible interpretations as the limit of sequences. Tradition has selected one of them to be the standard.