How to find the right b_n for limit comparsion test [closed]

manpreet Best Answer 2 years ago

so we had this question in our homework test:

Find infαinfα for which the series ∑∞n=1(2n)!n!(αn)n∑n=1∞(2n)!n!(αn)n converges

So I used the ratio test, and got that the limit is 4αe4αe. And by the ratio test, when α>4eα>4e the series converges.

But I wonder what happens when α=4eα=4e, even though it isn't part of the question. according to WolframAlpha the series diverges by the limit test, but I couldn't figure which bnbn to use.

I should note that Stirling approximation isn't on the syllabus, so I can't use that.

Thanks