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Math Help - help on taylor series (integral)

  1. #1
    Member
    Joined
    Jun 2008
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    105

    help on taylor series (integral)

    f(x) = integral from 0 to x
    ((cos(t)-1)/(t^2))

    I took the known taylor series of cos(t)
    = Sigma n=0 to infinity
    ((-1)^n t^2n))
    2n! t^2

    and moved the t^2 to the t exponential
    ((-1)^n t^2n-2))
    2n!

    What would the next step be?
    If I take the integral...
    ((-1)^n x^2n-1))
    2n!(2n-1)
    from n=1 to infinity

    Is this correct?
    Thanks.
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  2. #2
    Grand Panjandrum
    Joined
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    Thanks
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    Quote Originally Posted by khuezy View Post
    f(x) = integral from 0 to x
    ((cos(t)-1)/(t^2))

    I took the known taylor series of cos(t)
    = Sigma n=0 to infinity
    ((-1)^n t^2n))
    2n! t^2

    and moved the t^2 to the t exponential
    ((-1)^n t^2n-2))
    2n!

    What would the next step be?
    If I take the integral...
    ((-1)^n x^2n-1))
    2n!(2n-1)
    from n=1 to infinity

    Is this correct?
    Thanks.
    Integrating:

    \sum_{n=0}^{\infty}(-1)^n \frac{t^{2n-2}}{(2n)!}

    term by term gives:

    \sum_{n=0}^{\infty}(-1)^n \frac{t^{2n-1}}{(2n)!(2n-1)}

    so that looks like a yes (other than for the lower limit of summation.

    CB
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