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Continuous and differentiable Of Cos

by kala
Tags: continuous, differentiable
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kala
#1
Dec2-08, 04:38 PM
P: 21
1. The problem statement, all variables and given/known data
how could i prove that cos x= sum (n=1 to 00) [((-1)^n) * x^(2n)/((2n)!)]
is continuous and differentiable at each x in R



2. Relevant equations
the Taylor Expansion of cosine is the given equation


3. The attempt at a solution
basically i need to prove that the Taylor expansion of cos is differentiable and continuous. I think i need to use the Weierstrass M-test, however i could not figure out what M_n was, is there a different way to go about this one, or any suggestions for M_n
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kala
#2
Dec2-08, 05:05 PM
P: 21
Could I choose M_n to be something like this:

Let x be any real number and let L be large enough that x is in [-L, L]. Then M_n=L^(2n)/(2n)! and then use the Ratio Test to show convergence the convergence of the sum of M_n.

I think I could use the Ratio Test here, but I'm not sure how.


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