MathNet 2016-part-03 — 184 problems (EN–VI) Question 148

Let $a \gt 0$ and let $(x_{n})_{n \in \mathbb{N}}$ be the sequence satisfying $x_{1} = \frac{1}{a}$, $x_{n} = \frac{x_{n-1}}{1 + a n x_{n-1}}$ for all $n \geq 2$. Compute $\lim_{n \rightarrow \infty} (x_{1} + x_{2} + \ldots + x_{n})$.