How can I prove that this sequence is monotonic?

Question

So I'm supposed to prove that #{3^n/((2n)!)} _(n=1)^oo# is monotonic. From tinkering, I think it is indeed monotonic (decreasing), and I think I can prove it by induction. However, there's a method used by my professor that kinda looks like (I'll comment it below, the post is not permitting me to include another equation).

#a_(n+1)/a_n#

I get #3/(2n+1)# when I do that.

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Answer 1 · 2018-04-13T17:10:12

In this case the ratio #a_(n+1)/a_n<1# - and hence the sequence is monotonically decreasing.

#a_(n+1)/a_n = 3^(n+1)/((2(n+1))!) times ((2n)!)/3^n = 3/((2n+1)(2n+2))#

Now, for #n >= 1#, we have

#(2n+1)(2n+2) > 3#

and so

#a_(n+1)/a_n <1#

Since all the #a_n#s are positive, this implies that

#a_(n+1) < a_n#

and so the sequence is monotonically decreasing.