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 $a_{3}= \frac{4}{5} * \frac{104}{105} * \frac{304}{305}$)?
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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?
YiFan Tey
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I realized that it decreases (I calculated out the first few numbers), but I was unable to think of a way to show that this converges. – 324 Sep 25 '19 at 00:55
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@324 this actually proves that it decreases, not just a hunch. Do you know what the monotone convergence theorem is? If so, then try to show that your sequence is bounded. The fact that it decreases means it is monotone, and if it is bounded you are done. – YiFan Tey Sep 25 '19 at 00:57
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I am not familiar with it, but I will do some research on it. Thanks for the guidance. – 324 Sep 25 '19 at 01:00
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@324 No problem. It should be quite easy to obtain the bound $0<a_n\leq4/5$. – YiFan Tey Sep 25 '19 at 01:01