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Math Help - [SOLVED] Big O for functions of epsilon

  1. #1
    acseng
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    [SOLVED] Big O for functions of epsilon

    I'm needing a direction for this problem:

    show terms correct to O (\epsilon^2) for \frac{1+\epsilon}{1-\epsilon} , (1+\epsilon+\epsilon^2)^{1/2} , and sin(1+\epsilon).

    I'm thinking a series expansion for each expression. But I'm just a little confused. Any help or suggestions would be greatly appreciated.

    Thanks!
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  2. #2
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    Quote Originally Posted by acseng View Post
    I'm needing a direction for this problem:

    show terms correct to O (\epsilon^2) for \frac{1+\epsilon}{1-\epsilon} , (1+\epsilon+\epsilon^2)^{1/2} , and sin(1+\epsilon).

    I'm thinking a series expansion for each expression. But I'm just a little confused. Any help or suggestions would be greatly appreciated.

    Thanks!
    You need a series for each - up to the \epsilon^2 term.

    For the first, I'd first re-write it as \frac{2}{1 - \epsilon} - 1.

    For the second, I'd first re-write it as \left( \left[ \epsilon + \frac{1}{2} \right]^2 + \frac{3}{4} \right)^{1/2}.

    For the first, after the re-write I'd use the formula for an infinite geometric series.

    For the second (after the re-write) and third, I'd first get the standard expansions for (x^2 + a)^{1/2} and \sin x and then make the appropriate substitutions ......
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