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Alternating Series estimation theorem vs taylor remainder

  1. Nov 15, 2013 #1
    1. The problem statement, all variables and given/known data

    Let Tn(x) be the degree n polynomial of the function sin x at a=0. Suppose you approx f(x) by Tn(x) if abs(x)<=1, how many terms are need (what is n) to obtain an error less than 1/120

    2. Relevant equations

    sin(x)=sum from 0 to ∞ of (-1)^n*x^(2n+1)/(2n+1)!

    3. The attempt at a solution
    For the alternating series test, I plugged in 1 for x in the sin x macluarin series and got 1/(2(n+1)+1)! <= 1/120 and got n=1
    Then I used the Taylor's remainder theorem, got 1/(n+1)! <= 1/120 so n=4. why am I getting a different answer from Alternating series estimation theorem?
    Last edited: Nov 15, 2013
  2. jcsd
  3. Nov 15, 2013 #2


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    Let's write out a few terms of the two series. The alternating series is$$
    x -\frac{x^3} 3 +\frac {x^5} 5~...$$
    The Taylor series is$$
    0+x+0-\frac {x^3} 3 + 0 +\frac{x^5} 5~...$$Does that give you an idea?
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