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Taylor series

  1. Apr 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Let f be a function with derivatives of all orders and for which f(2)=7. When n is odd, the nth derivative of f at x=2 is 0. When n is even and n=>2, the nth derivative of f at x=2 is given by f(n) (2)= (n-1)!/3n

    a. Write the sixth-degree Taylor polynomial for f about x=2.
    b. In the Taylor series for f about x=2, what is the coefficient of (x-2)(2n) for n =>1 ?
    c. Find the interval of convergence of the Taylor series for f about x=2. Show the work that leads to your answer


    2. Relevant equations

    Taylor series

    3. The attempt at a solution

    a. I got the sixth-degree series.
    b. Will it just be (n-1)!/3n / (2n)! ??
    c. So will the Taylor series be [tex]\sum[/tex] (n-1)/3n * (x-2)^2n from n=1 to infinity ??
     
  2. jcsd
  3. Apr 8, 2009 #2
    No. Try again.

    No. See part b.
     
  4. Apr 8, 2009 #3
    Can you give me some hints ?
     
  5. Apr 8, 2009 #4
    You made a very simple mistake with the n's. Looking at some examples should reveal them. Try 2n = 2 and 2n = 4.
     
  6. Apr 8, 2009 #5
    For 2n=2, the coefficient will be (n-1)!/6 right ?

    So should it be (n-1)!/3^n /n! ??
     
  7. Apr 8, 2009 #6
    No. Try again and please show your work.
     
  8. Apr 8, 2009 #7
    So for n> or =1, 2n is always even so for the coeffecient we have to use the formula that is given: (n-1)!/3^n but also the denominator contains an even factorial so it will be 2n

    so will the coefficent be (n-1)!/3^n / 2n!
     
  9. Apr 8, 2009 #8
    Still wrong. If n is even, the coefficient of the (x-2)n in the Taylor series is f(n)(2)/n! = (n-1)!/(3nn!), so when you substitute n for 2k (I will use k instead of n because I believe this is what is confusing you), what do you get?
     
  10. Apr 9, 2009 #9
    So the coefficient will be (n-1)!/(3^n*n!) ??
     
  11. Apr 9, 2009 #10
    You keep answering the question (b) that could have been asked, instead of the question (b) that actually was asked.

    The question that could have been asked is this: what is the coefficient of (x-2)^n when n is even?

    By the way, what is 99!/100! ?
     
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