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Math Help - Simple Taylor series problem

  1. #1
    Newbie usvn's Avatar
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    Simple Taylor series problem

    Hi i just got introduced to Taylor and Maclaurin series in my calculus class and it was a very short explanation at the end of class. When i got to do doing my homework i found i had no idea where to begin, can someone help? i got the base idea but what to do is beyond me

    Find Taylor series for f(x) centered at the given value of a[assume that f has power series expansion. do not show that r_n_(x) goes to 0]

    f(x) = 1 + x + x^2
    Centered at a = 2
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  2. #2
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    Quote Originally Posted by usvn View Post
    Hi i just got introduced to Taylor and Maclaurin series in my calculus class and it was a very short explanation at the end of class. When i got to do doing my homework i found i had no idea where to begin, can someone help? i got the base idea but what to do is beyond me

    Find Taylor series for f(x) centered at the given value of a[assume that f has power series expansion. do not show that r_n_(x) goes to 0]

    f(x) = 1 + x + x^2
    Centered at a = 2
    f(2) = 7.

    f'(2) = 5.

    f''(2) = 2

    All higher derivatives are zero.

    So your series is f(x) = f(2) + (x - 2) f'(2) + (x - 2)^2 \frac{f''(2)}{2!} = \, ....

    You can check the final answer by expanding it all out and simplifying - you should get 1 + x + x^2 \, ....
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  3. #3
    Newbie usvn's Avatar
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    Thanks a lot
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  4. #4
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    Knowing that this is a polynomial of degee 2, so that its "Taylor series", about any point, is just another polynomial of degree 2, you could also write u= x- 2 so that x= u+2:

    f(x)= 1+ x+ x^2= 1+ (u+2)+ (u+2)^2= 1+ u+ 2+ u^2+ 4u+ 4
    = u^2+ 5u+ 7= (x- 2)^2+ 5(x- 2)+ 7
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