What is the closed form for this?

[Sam_]

Infinite Sum [n=1] 1/Fibonacci[n]

Or this,

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

Using some known mathematical constants.

 

[Gib Z]

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:
$$\psi = \sum_{k=1}^{\infty} \frac{1}{F_k} = 3.359885666243 \dots$$, 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.

 

[Sam_]

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.

 

[Gib Z]

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 =]

 

[Gib Z]

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:

$$\sum_{n=1}^{\infty} \frac{1}{n^n} = \int^1_0 x^{-x} dx$$