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What conditions must be satisfied by a function f to have a Taylor series centered at​ a? Choose the correct answer below. A. The function must be infinitely differentiable for all x in its domain. B. The series representation of the function f must converge to f on some open interval containing a. If it​ converges, that series is known as the Taylor series of f. C. The function f must have derivatives of all orders on some interval containing the point a. D. The remainder term of the​ nth-order Taylor polynomial Upper R Subscript n Baseline (x )Rn(x) must approach 0 as n approaches infinity.

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jmonterrozar

Answer:

D. The remainder term of the​ nth-order Taylor polynomial Upper R Subscript n Baseline (x )Rn(x) must approach 0 as n approaches infinity.

Step-by-step explanation:

We have that a Taylor series has the following form:

[tex]f (x) = f(a) + \frac{f'(a)}{1!}*(x-a)+ \frac{f''(a)}{2!}*(x-a)^{2}+ \frac{f'''(a)}{3!}*(x-a)^{3} ...[/tex]

Therefore, as a result of this we can say that the correct answer is D.

Because the function f must have derivatives of any order in some interval that contains the point a Taylor Series of the function f (x) = f (a) + f ^ prime (a) / 1! (x - a) + f prime / 2! (x - a) ^ 2 + ft ^ prime (a) / 3! (x - a) ^ 3 from above we can function f must have derivatives of every order in some interval that contains point a.

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