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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
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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.
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September 4th, 2018, 12:21 PM   #3
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Or just $\displaystyle \text{QPochhammer}\small\left[ \frac 1 2\right]$.
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September 10th, 2018, 09:54 PM   #4
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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.
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