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Ratio Test

What is the conclusion of the ratio test when applied to Σ,∞,n=1  (n!)^2/(3n)!

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Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
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Look at the ratio of successive terms:
 
((n+1)!²/(3n+3)!)/(n!²/(3n)!)
=((n+1)!²/n!²)/((3n+3)!/(3n)!)
=(n+1)²/((3n+3)(3n+2)(3n+1))
The highest order terms in the top and bottom are n²/(27n³)
In the limit as n→∞, the ratio goes to zero, so the series converges.