<s>[INST] Convergence of product of rational functions How can I prove that the sequence $a_{n} = \frac{4}{5} * \frac{104}{105} * ... * \frac{50n^{2} - 50n + 4}{50n^{2} - 50n + 5}$ converges or not (where $a_{1} = \frac{4}{5}$, $a_{2} = \frac{4}{5} * \frac{104}{105}$, and $a_{3}= \frac{4}{5} * \frac{104}{105} * \frac{304}{305}$)?
 [/INST] Hint: Your sequence is
$$a_N=\prod_{n=0}^N\frac{50n^2-50n+4}{50n^2-50n+5}=\prod_{n=0}^N\left(1-\frac1{50n^2-50n+5}\right).$$
Each additional term in the product causes it to decrease in value since the $a_n$ are positive and the term is less than $1$. Now can you think of a way to use monotone convergence theorem?
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