Where is the recurrence stable in the (n,x) plane for increasing n?

[member 428835]

Homework Statement

In the $(n,x)$ plane, where is the recurrence stable for increasing $n$?

$$E_{n+1}(x) = \frac{1}{n}\left( \exp(-x)-xE_n(x)\right):E_n\equiv \int_1^\infty \frac{\exp(-xt)}{t^n}\,dt$$
and $n\in \mathbb{N},\:x\geq0$.

Homework Equations

Nothing comes to mind.

The Attempt at a Solution

I really don't know how to proceed because I don't understand what "stable" means in this context. I was hoping someone with more experience could help me out. I typically have an approach when posting here, but this time I'm stuck.

Nothing on this topic is in the book we're using, so I'm stuck.

 

[Mark44]

"Stable" usually means unchanging, so my best guess in this problem is that for what x will $E_n = E_{n + 1}$?

 

[member 428835]

Ohhhh gotcha! And thanks! So my thoughts are to write a code that computes the difference between $E_n$ and $E_{n+1}$ and see when that difference is very small. What do you think?