How do you use the ratio test to test the convergence of the series #sum_(n=1)^oo (n!)/((2n+1)!)#?

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Answer 1 · 2018-05-19T06:53:17

The series converges

Let #u_n=(n!)/((2n+1)!)#

Then, the ratio test is

#|a_(n+1)/a_(n)|=|(((n+1)!)/((2(n+1)+1)!))/((n!)/((2n+1)!))|#

#=|((n+1)!)/(n!)((2n+1)!)/((2n+3)!)|#

#=|(n+1)/((2n+2)(2n+3))|#

#=|1/(2(2n+3))|#

#1/(2(2n+3))>0# as #n in [1, +oo)#

Therefore,

#lim_(n->oo)|1/(2(2n+3))|=lim_(n->oo)1/(2(2n+3))#

#=0#

As the limit is #<1#,by the ratio test, the series converges