Hello, dhiab!
Ratio Test: .
. . . . . . . .
Divide top and bottom by
Then: .
Since , the series diverges.
Soroban,
I think you made an error in your ratio test: since ,
then
,
which prevents the cancelling you did.
Dhiab, the key to this limit is Stirling's approximation:
for large n.
--Kevin C.