I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit.
an = (-2)n / (n!)
In solving this problem, I am not allowed to use any form or variation of the Ratio Test.
2. The attempt at a solution
I was thinking that I could reasonably compare the above sequence to bn = (1/2)(-2)n/(3n-2), like so:
(1/2)(-2)n/(3n-2) <= (-2)n / (n!) <= (1/2)(2)n/(3n-2)
By manipulating the bn statement, I can make a geometric sequence:
bn = (1/2)(-2)2(-2/3)n-2
I know this converges since r = -2/3, and the absolute value of r is less than 1. Taking the limit of the sequence (bn) as n goes to infinity, I find that the value of the sequence approaches zero. Then, since both bn and its absolute value converge to zero, I can say that an converges and its limit is zero as well, by the Squeeze Theorem.
I honestly have no idea if this is right or not. My professor gave the class this problem as part of a practice test, and we weren't given an answer key. The closest problems I can find in my textbook all use the Ratio Test in their solutions, so they're no help. The corresponding test is on Monday, and we will not be going over any solutions to the practice test beforehand, so I'd really like to make sure my thought process is correct.