# How do you find the power series representation for the function #f(x)=cos(2x)# ?

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To find the power series representation for the function ( f(x) = \cos(2x) ), we can use the Maclaurin series expansion of ( \cos(x) ) and then substitute ( 2x ) for ( x ):

[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} ]

Substituting ( 2x ) for ( x ) gives:

[ \cos(2x) = \sum_{n=0}^{\infty} \frac{(-1)^n (2x)^{2n}}{(2n)!} ]

Simplify the expression:

[ \cos(2x) = \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n} x^{2n}}{(2n)!} ]

[ \cos(2x) = \sum_{n=0}^{\infty} \frac{(-1)^n 4^n x^{2n}}{(2n)!} ]

Therefore, the power series representation for ( f(x) = \cos(2x) ) is:

[ f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n 4^n x^{2n}}{(2n)!} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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