What is the closed form for this?

In summary, the first sum is known as the Reciprocal Fibonacci Constant, which is an irrational number and its exact value is not expressible in terms of common constants. The second sum also has no closed form solution, but it can be represented by the integral of x to the power of negative x from 0 to 1.
  • #1
Sam_
15
0
Infinite Sum [n=1] 1/Fibonacci[n]

Or this,

Infinite Sum[n=1] 1/(n^n)

Using some known mathematical constants.
 
Mathematics news on Phys.org
  • #2
Welcome to Physics forums sam! Perhaps you don't exactly know, or you just want to give us problems we can't do to torture us, but:

The first sum is equal to a known mathematical constant, and its called the Reciprocal Fibonacci Constant:
[tex]\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots[/tex], which has been proven to be irrational, but it is currently not known if that constant is expressible in terms of more elementary or common constants.

The second One I know converges, though I have never seen a closed form solution for.
 
  • #3
Hi Gib_Z,

Sorry, I didn't mean to torture you

And thanks for the explanation. I was wondering about this the other day and thought what better place to ask about it than here.
 
  • #4
This place is about the best place to ask in my opinion as well =] You are really quite lucky, if I hadn't read through the wikipedia article on the Fibonacci numbers just last week, I would not have even remembered about that constant =]
 
  • #5
Doing some crawling on the internet I have found out for sure that there is no closed form for the second series, though there is a nice identity that I am attempting at this moment to prove, the derivation isn't given...I should have it by tomorrow, but here's the identity now:

[tex]\sum_{n=1}^{\infty} \frac{1}{n^n} = \int^1_0 x^{-x} dx[/tex]
 

What is the closed form for this?

The closed form refers to an equation or formula that describes a mathematical relationship between variables in a concise and explicit manner, without the use of summations or limits.

Why is it important to find the closed form?

Finding the closed form allows for a deeper understanding of the underlying mathematical relationship between variables, making it easier to analyze and manipulate the equation. It also allows for easier and more efficient computation and the ability to make predictions and generalizations.

How do you determine the closed form for a given equation or sequence?

The process for determining the closed form varies depending on the type of equation or sequence. Generally, it involves identifying patterns and using algebraic manipulations or techniques such as induction or finite differences to simplify the expression.

Can every equation or sequence have a closed form?

No, not every equation or sequence has a closed form. Some mathematical relationships are too complex to be expressed in a closed form or do not follow any discernible pattern. In these cases, approximations or numerical methods may be used to find a solution.

Are there any disadvantages to using the closed form?

While the closed form can provide a more elegant and concise representation of a mathematical relationship, it may not always be the most practical or efficient solution. In some cases, using the closed form may result in a loss of precision or introduce errors. It is important to consider the specific context and purpose of the equation before deciding to use the closed form.

Similar threads

  • General Math
Replies
1
Views
179
  • General Math
Replies
7
Views
1K
  • General Math
Replies
2
Views
1K
  • General Math
Replies
5
Views
2K
  • General Math
Replies
9
Views
1K
Replies
15
Views
1K
  • General Math
Replies
1
Views
2K
  • General Math
Replies
6
Views
804
  • General Math
Replies
1
Views
939
Replies
4
Views
285
Back
Top