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Math Help - Taylor polynomial approximation

  1. #1
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    Taylor polynomial approximation

    e^{-5} \approx \sum \frac{(-5)^i}{i!} = \sum \frac{(-1)^i 5^i}{i!}
    i runs from 0 to 9
    e^{-5}= \frac{1}{e^5} \approx \frac{1}{\sum\frac{5^i}{i!}}
    i runs from 0 to 9

    Sorry, I don't know how to put the indices on the summation. The question is which formula gives the most accuracy comparing to the actual value of e^{-5} correct to three digits, which is 6.74 x 10^{-3}.
    I computed using both formulas and figured out that the second formula gives the most accuracy, but I don't have a solid answer for why it is more accurate than the other.
    I guess it's is because in the second formula we write down the exact formula for e^{-5}, then use the approximation after. On the first formula, we approximate first.
    Anyone can give me some thought on this. Thank you.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by namelessguy View Post
    e^{-5} \approx \sum \frac{(-5)^i}{i!} = \sum \frac{(-1)^i 5^i}{i!}
    i runs from 0 to 9
    e^{-5}= \frac{1}{e^5} \approx \frac{1}{\sum\frac{5^i}{i!}}
    i runs from 0 to 9

    Sorry, I don't know how to put the indices on the summation. The question is which formula gives the most accuracy comparing to the actual value of e^{-5} correct to three digits, which is 6.74 x 10^{-3}.
    I computed using both formulas and figured out that the second formula gives the most accuracy, but I don't have a solid answer for why it is more accurate than the other.
    I guess it's is because in the second formula we write down the exact formula for e^{-5}, then use the approximation after. On the first formula, we approximate first.
    Anyone can give me some thought on this. Thank you.
    Look at the form of the remainder for the two expansions, and what happens to the remainder in the second when you take a reciprical.

    RonL
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Hello,
    Quote Originally Posted by namelessguy View Post
    Sorry, I don't know how to put the indices on the summation.
    [tex]\sum_{i=0}^{9}[/tex] gives \sum_{i=0}^{9}.
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