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Showing that a series diverges 
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#1
Nov1309, 07:15 PM

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1. The problem statement, all variables and given/known data
Show that the series [tex]\sum_{0}^{\infty}(2n)!/(n!)^2*(1/4)^n[/tex] diverges 2. Relevant equations I don't know which convergence test to use 3. The attempt at a solution I don't have one, because I don't know which convergence test to use. If someone can tell me what to use, I will be able to figure out this problem. 


#3
Nov1309, 08:01 PM

P: 6

I know that the basic ones such as ratio, divergence, alternating series, comparison, and integral don't work with this series. I believe it uses a test that I haven't learned about, so I was wondering what that could be.



#4
Nov1309, 08:12 PM

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Showing that a series diverges
I am just leaving for the evening so I don't have time to work on it myself. But I would be very surprised if the ratio test won't settle it. Did you try the root test?
I will check back later. 


#5
Nov1309, 08:42 PM

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Show us what you've done... 


#6
Nov1409, 12:54 PM

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Clarification: Is that [itex](1/4)^n[/itex] in the numerator or the denominator of the fraction? I'm guessing the numerator, making the [itex]4^n[/itex] in the denominator, which makes the problem tougher than I thought at first glance. Is that right? I'm getting the ratio test fails too...



#7
Nov1409, 12:59 PM

P: 6

It's just the first fraction times (1/4)^{n}. I guess there should be two parenthesis around the first fraction so it's just that quantity multiplied by the (1/4)^{n}



#8
Nov1409, 02:37 PM

P: 1,622

My initial response was deleted by the moderators, but you can prove that the sum diverges by considering the sequence [itex]a_n = 1/n[/itex].



#9
Nov1409, 03:12 PM

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You can also learn things about series like this using Stirling's approximation.



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