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# Simplify factorials

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3
+37

Hey guys,

I'm trying to use the ratio test to find divergence/convergence, however I was never taught factorials in depth so I don't understand how they work.

I was given the question

$$\sum_{n=1}^{inf}\frac{3^nn!n!}{(2n)!}$$

The way that I did it the expression diverges because it goes to infinity but that's apparently wrong, so how would I be able to solve something like this?

bingby  Apr 18, 2017
#1
+37
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I tried to simplify it and was getting the limit as n goes to infinity of 3(n+1)^2 divided by (2n+2)(2n+1). The numerator and denominator have the same degree of n so they cancel out and the limit as n goes to infinity reaches the constant 3. 3>1 so from my calculations it should still diverge but I am still wrong.

bingby  Apr 18, 2017
#3
+27219
+1

As n tends to infinity (n + 1) tends first to n, and (2n + 1) tends to 2n and (2n + 2) tends to 2n, so (n+1)^2/(2n+2)(2n+1) tends to n^2/(2n)^2 = 1/4.

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Alan  Apr 18, 2017
#2
+27219
+1

Perhaps this will help:

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Alan  Apr 18, 2017