# Is it possible to convert any recursive definition into a non-recursive one?

**The context is sequences and series. That is, I have an infinite series which has terms that are defined recursively. I can't a non-recursive definition...
The problem:
The nth term of the infinite series A is denoted by A(n) and the finite sum of A up to the nth term is denoted by S(n).
A is defined by A(k+1) = A(k)/S(k), with A(1) = 1 and S(1) = 1.
I worked out that for k > 1:
A(k) = 1/[S(k-1)*S(k-2)...S(1)]
S(k) = 1/[S(k-1)*S(k-2)...S(1)] + 1/[S(k-2)*S(k-3)...S(1)] + 1/[S(k-3)*S(k-4)...S(1)] + ...+ 1/S(1)
The expansion is so complicated, I don't think it's even possible to express it in a contained form. I'm wondering if there's some method I don't know about that can produce a non-recursive formula for the terms in this series.
**

TELL US , if you have any answer