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Math Help - Evaluating an indefinite integral as an infinite series

  1. #1
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    Evaluating an indefinite integral as an infinite series

    Greetings! I'm not sure how to approach this question, it's asking that I rewrite this:

    (sin(x)-x)/x^3 dx

    as an infinite series. Any help is appreciated!
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  2. #2
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    Re: Evaluating an indefinite integral as an infinite series

    Write sin(x) as a Taylor's series. Then you can subtract x, divide by x^3 and integrate term by term.
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    Re: Evaluating an indefinite integral as an infinite series

    Ok let me see...
    sinx as a Taylor series would be the sum of ((-1)^n/(2n+1)!)*x^(2n+1)

    so then would we have (((-1)^n/(2n+1)!)*x^(2n+1)/x^3) - (x/x^3)?
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    Re: Evaluating an indefinite integral as an infinite series

    Yes, and surely you can now simplify...
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    Re: Evaluating an indefinite integral as an infinite series

    It might make more sense if you write it out as sin(x)= x- x^3/3!+ x^5/5!- x^7/7!+ x^9/9!- \cdot\cdot\cdot.
    Then sin(x)- x= - x^3/3!+ x^5/5!- x^7/7!+ x^9/9!- \cdot\cdot\cdot
    and \frac{sin(x)- x}{x^3}= -1+ x^2/5!- x^4/7!+ x^6/9!- \cdot\cdot\cdot
    Integrate that.
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    Re: Evaluating an indefinite integral as an infinite series

    Ah! Alright that makes much more sense now. Thank you very much!
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