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