How do you take the limit of n->infinity x^(2n+2)2^(2n+1)/(2n+2)!?

original question was to use the Lagrange form of the remainder to prove that the series converges to (cos(x))^2. I found that the Maclaurin series was x^2n2^(2n+1)/(2n)! then i used the ratio test to prove convergence. now i need to prove that the limit of the lagrange remainder is 0, which will show that it... show more original question was to use the Lagrange form of the remainder to prove that the series converges to (cos(x))^2. I found that the Maclaurin series was x^2n2^(2n+1)/(2n)! then i used the ratio test to prove convergence. now i need to prove that the limit of the lagrange remainder is 0, which will show that it converges to the given function. I'm just not sure how to prove that limit, or if i have the right stuff for the remainder. So if you like a good math problem, I'd appreciate some help!
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