Find a general expression for $\frac{p}{p+1 - \frac{p}{p+1 - \frac{p}{p+1 - \ldots}}}$ $n$ times for any value of $p \in \Bbb R$ .

Question

Find a general expression for $\frac{p}{p+1 - \frac{p}{p+1 - \frac{p}{p+1 - \ldots}}}$ $n$ times for any value of $p \in \Bbb R$ .

Obs: Consider $n=1 : \frac {p}{p+1}$ and $n=2: \frac {p}{p+1 - \frac{p}{p+1}}$

I don't know how to work with continued fractions because i never worked with them before.

Any hints?

Answer 1

In case you desire an answer to check against, I will leave one below. Be sure to solve the problem before looking at what I've written!

As mentioned by Somos, we can define each term of this series by $$a_{n+1} = p/(p+1-a_n)\ \ \text{ with } \ \ a_0 = 0$$ The first few terms of this series are as follows: $$a_1 = \frac{p}{p+1}$$ $$a_2 = \frac{p}{p+1 - \frac{p}{p+1}} = \frac{p(p+1)}{p(p+1) + (p+1) - p} = \frac{p^2+p}{p^2+p + 1}$$ $$a_3 = \frac{p}{p+1 - \frac{p^2+p}{p^2+p + 1}} = \frac{p(p^2+p+1)}{p(p^2+p+1) + (p^2+p+1) - (p^2+p)} = \frac{p^3+p^2+p}{p^3+p^2+p+1}$$ At this point it should be clear that the $n^{th}$ term of the sequence will be: $$a_n = \frac{\sum_{i=1}^n p^{i}}{\sum_{i=0}^n p^{i}}$$