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Math Help - antiderivative of a power series

  1. #1
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    antiderivative of a power series

    Hi everyone. My final is tomorrow, and I'm going over a lot of stuff that plans to be covered on the test. My professor had to leave town suddenly before he could fully cover everything, but chose not to change the test. He never went over differentiation or antiderivatives of a power series. I was wondering if anyone could help me out with both of these things for this given problem, and Im sure I could apply these techniques to all future problems.

    Differentiation and antiderivative of the following power series:


    this would be a great help! thanks so much!
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    Quote Originally Posted by etha View Post
    Hi everyone. My final is tomorrow, and I'm going over a lot of stuff that plans to be covered on the test. My professor had to leave town suddenly before he could fully cover everything, but chose not to change the test. He never went over differentiation or antiderivatives of a power series. I was wondering if anyone could help me out with both of these things for this given problem, and Im sure I could apply these techniques to all future problems.

    Differentiation and antiderivative of the following power series:


    this would be a great help! thanks so much!
    you have the formula \frac {2n}{2n + 1} \cdot x^n ............ \frac {2n}{2n + 1} is a constant. you can integrate and differentiate using the power rules
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    Quote Originally Posted by etha View Post
    Hi everyone. My final is tomorrow, and I'm going over a lot of stuff that plans to be covered on the test. My professor had to leave town suddenly before he could fully cover everything, but chose not to change the test. He never went over differentiation or antiderivatives of a power series. I was wondering if anyone could help me out with both of these things for this given problem, and Im sure I could apply these techniques to all future problems.

    Differentiation and antiderivative of the following power series:


    this would be a great help! thanks so much!
    Power series integrate and differentiate term wise (like a polynomial)

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

    Now for the integral

    \int \sum_{n=0}^{\infty}\frac{2nx^n}{2n+1}= C+\sum_{n=0}^{\infty}\frac{2nx^{n+1}}{(2n+1)(n+1)}  =
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    what exactly are the power rules? When I mean my professor skipped town, he literally skipped town a week before finals and expected us to learn the material on our own. The book I have is really confusing, and the help from this forum has much easier to understand.
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    Quote Originally Posted by TheEmptySet View Post
    Power series integrate and differentiate term wise (like a polynomial)

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

    Now for the integral

    \int \sum_{n=0}^{\infty}\frac{2nx^n}{2n+1}= C+\sum_{n=0}^{\infty}\frac{2nx^{n+1}}{(2n+1)(n+1)}  =
    i see, that helps a lot. I have a question though, when you differentiate, why does the 2n multiply by n and become n^2
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by etha View Post
    what exactly are the power rules? When I mean my professor skipped town, he literally skipped town a week before finals and expected us to learn the material on our own. The book I have is really confusing, and the help from this forum has much easier to understand.
    As you may have figured out from EmptySet's post...

    The Power Rules for differentiation and integration are respectively as follows:

    \frac d{dx} x^n = n x^{n - 1}


    \int x^n~dx = \frac {x^{n + 1}}{n + 1} + C for n \ne -1. (if n = -1, \int x^{-1}~dx = \ln x + C)
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    Quote Originally Posted by etha View Post
    i see, that helps a lot. I have a question though, when you differentiate, why does the 2n multiply by n and become n^2
    n \cdot n = n^2 because x^a \cdot x^b = x^{a + b}
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    Behold, the power of SARDINES!
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    Quote Originally Posted by etha View Post
    i see, that helps a lot. I have a question though, when you differentiate, why does the 2n multiply by n and become n^2
    It is just simplification

    2n(n)=2n^2 n is just a parameter.

    this is just like 2x(x)=2x^2

    Good luck.

    Edit: Dang you are speedy.
    I need to get faster
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by TheEmptySet View Post
    Edit: Dang you are speedy.
    I need to get faster
    Hehe, i'm not nearly fast enough to be around here. I was just lucky. anyway, i'll leave you to run the forum, i'm logging off
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    Quote Originally Posted by TheEmptySet View Post

    2n(n)=2n^2 n is just a parameter.

    this is just like 2x(x)=2x^2
    well i know that when you multiply the n in it becomes n^2 lol. I just don't understand why you mutiply it by n. I'm sorry if it seems like I'm a slow learner!
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  11. #11
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    Quote Originally Posted by etha View Post
    well i know that when you multiply the n in it becomes n^2 lol. I just don't understand why you mutiply it by n. I'm sorry if it seems like I'm a slow learner!
    It is what the power rule says we should do. see post #6
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