My Math Forum Convergence of sequence

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September 4th, 2018, 11:50 AM   #1
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Convergence of sequence

Is this sequence convergent?
I rewrited this product so it looks simpler, so, in this fraction, I see the pattern by which the factors behave (the n-th factor in the numerator is the n-th factor in the denominator -1), but I don't have idea how to calculate that value, and determine if this sequence converges? Any help appreciated!
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 September 4th, 2018, 11:59 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,100 Thanks: 1093 Mathematica is calling the limit as $n \to \infty ~~\text{QPochhammer}\left[ \dfrac 1 2, \dfrac 1 2\right]$ It has a numeric value of approximately $0.2887881$. Last edited by skipjack; September 4th, 2018 at 12:03 PM.
 September 4th, 2018, 12:21 PM #3 Global Moderator   Joined: Dec 2006 Posts: 19,541 Thanks: 1750 Or just $\displaystyle \text{QPochhammer}\small\left[ \frac 1 2\right]$.
 September 10th, 2018, 09:54 PM #4 Math Team   Joined: Nov 2014 From: Australia Posts: 688 Thanks: 243 If you only want to know whether it converges, prove that it is decreasing and bounded below by 0. Then use the monotone convergence theorem.

 Tags convergence, divergence, limit, sequence

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