# An infinite continued fraction of $x^n$

I coded in SageMath to compute the 50^{th} convergent of $f(x)$ and plot for integers $x=1,\dots,50$:

```
def f(x, n_terms):
result = []
for i in range(n_terms):
result.extend([x**i, x**i])
return continued_fraction([0] + result).n()
list_plot(([f(x,50) for x in range(1,50)]), title='$f(x)=\\dfrac{x}{x+\\dfrac{x^2}{x^2+\\dfrac{x^3}{x^3+\\ddots}}}$')
```

My question is to make a continuous plot of $f(x)$.
I have difficulty in that `continued_fraction`

only support integer values of $x$.

Your function

`f`

cannot be evaluated at non-integer values`x`

since`continued_fraction()`

expects a list with integer elements. Hence,`f`

(as defined) is not a continuous function and cannot be plotted as such. Do you mean a continuous interpolation of`f`

from its values at integer points or something else?Yes, the built-in function

`continued_fraction()`

expects a list of integers, but I want to compute $$f(x)=\cfrac{x}{x+\cfrac{x^2}{x^2+\cfrac{x^3}{x^3+\cdots}}}$$ $f$ is a continuous function for real number $x>1$. Does SageMath have built-in function to compute continued fractions of real numbers?