What is the value of f^{n}(x) when n approaches infinity?

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In summary, the conversation discusses iterating a function defined by f(x) = 0.6x + 2100 and finding an elementary function for f^n(x). The general pattern for f^n(x) is found and simplified for n → ∞ to be 3500*(0.6/(1-0.6)) = 8750. There is some confusion about the constant value, as it is initially described as 2100 but later mentioned as 3500.
  • #1
Bipolarity
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Consider the function defined by
[tex] f(x) = 0.6x + 2100 [/tex]

Suppose this function is iterated [itex]n[/itex] times.

Express [itex] f^{n}(x) [/itex] as an elementary function of [itex]x[/itex].

The problem is actually a simplified version of a finance problem I am trying to solve. I am not sure there is a solution so if a solution does not exist, please help me prove it.

BiP
 
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  • #2
This is straightforward if you iterate a few times.
f2(x) = 0.6(0.6x + 2100) + 2100 = (0.6^2)x + 0.6*2100 + 2100
f3(x) = 0.6((0.6^2)x + 0.6*2100 + 2100) + 2100 = (0.6^3)x + (0.6^2)2100 + 0.6*2100 + 2100
This should be enough to see the general pattern for fn(x).
 
  • #3
Would this be correct?

[tex] f^{n}(x) = .6^{n}x + 3500(.6^{n-1} + .6^{n-2} + .6^{n-3} + ... + .6^{0}) [/tex] for [itex] n>1 [/itex]

BiP
 
  • #4
Yes, assuming you meant 2100 instead of 3500. It is a linear function of x, which satisfies the elementary function requirement. The constants are just rather large polynomials of 0.6.
 
  • #5
slider142 said:
Yes, assuming you meant 2100 instead of 3500. It is a linear function of x, which satisfies the elementary function requirement. The constants are just rather large polynomials of 0.6.

Interesting.
What if we make [itex] n → ∞ [/itex] ?

Can we simplify in that case?

My guess is that the answer will be [itex] 3500*(\frac{.6}{1-.6}) = 5250 [/itex].

BiP
 
  • #6
Bipolarity said:
Interesting.
What if we make [itex] n → ∞ [/itex] ?

Can we simplify in that case?

My guess is that the answer will be [itex] 3500*(\frac{.6}{1-.6}) = 5250 [/itex].

BiP

It should be 3500*(1/(1-0.6)) = 35000/4 = 8750 for the sum of the geometric series. Is 3500 the correct value? You have 2100 in its place in the first post.
 

Related to What is the value of f^{n}(x) when n approaches infinity?

What is a function?

A function is a block of code that performs a specific task. It takes in inputs (known as parameters) and returns an output. Functions are used to organize and modularize code, making it easier to read, understand, and reuse.

What is the difference between a named function and an anonymous function?

A named function is a function that is declared with a specific name, which can be used to call the function. An anonymous function, on the other hand, is a function that is not given a specific name and is usually declared as a variable or passed as an argument to another function.

What is the purpose of a return statement in a function?

A return statement is used to specify the value that the function will return when it is called. It allows us to use the output of a function in other parts of our code and is essential for functions to be able to perform a specific task and provide a result.

What is a higher-order function?

A higher-order function is a function that takes another function as an argument or returns a function as its output. This allows for more flexibility and functionality in our code, as we can pass different functions as arguments to the higher-order function and get different results.

How do I call a function in my code?

To call a function, you simply use the name of the function followed by parentheses. If the function requires any parameters, they should be passed within the parentheses. For example, if the function is named "addNumbers" and takes in two numbers as parameters, the function call would look like this: addNumbers(2, 3).

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