Computing the limit of the sequence : $ x_n = \prod_{i=1}^{n} 1 + a^{2i} $

Question

The given sequence is $$ x_n = \prod_{i=1}^{n} 1 + a^{2i} ~,~\mbox{ a} \in \mathbb{R}$$

I have to compute $\displaystyle \lim_{n \rightarrow \infty} x_n $.

I struggle finding something for $ a \in ]-1,1[ \setminus \{0\}.$ Do you have any idea ? For the other values, applying the criterea $ \frac{x_{n+1}}{x_n}$ is sufficient.

Answer 1

The fact is that when you develope the expression of $x_n$, you find this : $$ \sum_{k=0}^{2n} a^k $$ so then to compute the limit, you just have to compute the sum of a geometrical sequence. Then it does converge for value of $a$ between $ ]-1,1[ $.