How to sum up the series of x + 1/x + 1/(x + 1/x) + 1/(x + 1/x + 1/(x + 1/x)) for some finite number of terms?

Question

I'm having trouble in coming up with a formula for this series:

$$S = x + \frac1x + \cfrac1{x + \frac1x} + \cfrac1{x + \frac1x + \cfrac1{x + \frac1x}} +\cfrac1{x + \frac1x + \cfrac1{x + \frac1x} + \cfrac1{x + \frac1x + \cfrac1{x + \frac1x}} } + \cdots + a_n $$

How can I find $S$ ?

Thanks :)

Answer 1

You can define the function $f(x) = 1 + 1/x$ and then call it $n$ times

n=

1. $$f(x) = x+\frac 1 x$$
2. $$f(f(x)) = f(x) + \frac 1 {f(x)}$$
3. $$f(f(f(x))) = f(f(x)) + \frac 1 {f(f(x))}$$

etc.