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Math Help - A few problems encountered in exam revision

  1. #1
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    A few problems encountered in exam revision

    Hi. Any help with these would be much appreciated.

    1. By considering the infinite series (SUMMATION SIGN with infinity on top and n=0 on bottom) of x^n for x< 1 show that:

    (SUMMATION SIGN with infinity on top and n=0 on bottom) (n^2)(x^n) = x(1-x^2)/(1-x)^4.

    I know this is to do with differentiating the previous derivative (x/(1-x)^2), but i got in a mess with it.

    2.Find the indefinate integral of x^2*e^(x^3)

    This is integration by parts but the x^3 is confusing me as i don't know how to deal with it. Basically my question is; how do you integrate e^(x^3)?

    3. Use your calculator to find the value of the following expression when x=0.01 to 3dp:

    y(x) = (e^(2x) - 2(1 +2x)^(1/2) +1) / (cos (x/2) -1)

    The previous part of the question asked me to obtain Maclaurin series and re-write the above expression which i did. However when i substituted x=0.01 the 2 answers from the series and the exact did not match. I maybe approaching this question incorrectly. What is the meothod for this



    Again any help would be much appreciated.
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  2. #2
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    Quote Originally Posted by schteve View Post
    Hi. Any help with these would be much appreciated.

    1. By considering the infinite series (SUMMATION SIGN with infinity on top and n=0 on bottom) of x^n for x< 1 show that:

    (SUMMATION SIGN with infinity on top and n=0 on bottom) (n^2)(x^n) = x(1-x^2)/(1-x)^4.

    I know this is to do with differentiating the previous derivative (x/(1-x)^2), but i got in a mess with it.
    Hi

    This is what I would do

    \sum_{n=0}^{+\infty} x^n = \frac{1}{1-x}

    Differentiating with respect to x

    \sum_{n=0}^{+\infty} (n+1) x^n = \frac{1}{(1-x)^2}

    Differentiating again with respect to x

    \sum_{n=0}^{+\infty} (n+2)(n+1) x^n = \frac{2}{(1-x)^3}

    Using (n+2)(n+1) = n^2 + 3n + 2 = n^2 + 3(n+1) - 1 or n^2 = (n+2)(n+1) - 3(n+1) + 1 you are able to find

    \sum_{n=0}^{+\infty} n^2 x^n
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  3. #3
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    Quote Originally Posted by schteve View Post
    2.Find the indefinate integral of x^2*e^(x^3)

    This is integration by parts but the x^3 is confusing me as i don't know how to deal with it. Basically my question is; how do you integrate e^(x^3)?
    .
    No need for integration by parts.
    the substitution u=x^3 will solve it easily .
    By the way, It is not easy to evaluate \int e^{(x^3)} dx
    since it has unelementary functions.
    see this:
    integrate e^(x^3) - Wolfram|Alpha
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