September 4th, 2018, 12:50 PM  #1 
Newbie Joined: Mar 2018 From: Split, Croatia Posts: 13 Thanks: 0  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 nth factor in the numerator is the nth 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! 
September 4th, 2018, 12:59 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,174 Thanks: 1143 
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 01:03 PM. 
September 4th, 2018, 01:21 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,887 Thanks: 1836 
Or just $\displaystyle \text{QPochhammer}\small\left[ \frac 1 2\right]$.

September 10th, 2018, 10:54 PM  #4 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
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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