Q:

Limit of sequence without Taylor series

Consider the sequence $(x_n)_{n \in \mathbb N^*}, x_1 = 2$ with the recurrence formula:
$$\frac{1}{x_{n+1}^{\frac{1}{n+1}} - 1} - \frac{1}{x_{n}^{\frac{1}{n}} - 1} = 1$$
Evaluate $$\lim _{n \to \infty} x_n$$
I tried calculating some terms and it seems like  $$\lim _{n \to \infty} x_n = e$$
but I really don't know how to start. Any hint would be appreciated.

A:

HINT
To eliminate the inconvenient denominator powers, substitute $a_n = x_n^{1/n}$ to get the recurrence
$$
\frac{1}{a_{n+1}-1} - \frac{1}{a_n-1} = 1
$$
and now let
$$
b_n = \frac{1}{a_n-1}
$$
to get the linear recurrence
$$
b_{n+1} - b_n = 1
$$
which is solvable using ordinary linear methods. 

