A sum involving Fibonacci numbers, $\sum_{k=1}^\infty F_k/k!$

Question

Let $F_k$ be Fibonacci numbers. I am looking for a closed form of the sum $\sum_{k=1}^\infty F_k/k!$.

I tried to use Wolfram Alpha, but it is not doing the sum Fibonacci[k]/k! , k=1 to infinity.

Can someone tell what is the problem with WA and what this sum equals to?

Answer 1

Using Mma, Limit[Sum[Fibonacci[k]/k!, {k, 1, p}], p -> Infinity] gives an answer and
Sum[Fibonacci[k]/k!, {k, 1, Infinity}] gives the same. Is your problem with Wolfram Alpha ?

Answer 2

A hint:

You can prove for yourself or find in a book a formula of the form $${\rm fib}(k)=a\lambda^k + b\mu^k$$ with certain constants $a$, $b$, $\lambda$, $\mu$. The requested sum can then be easily written as a sum of two exponentials.