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Math Help - Convergence with ratio/root test and power series

  1. #1
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    Convergence with ratio/root test and power series

    Okay so i missed the days in class when we went over power series and ratio/root test because of another exam. Reading my book i cant figure out how to do ratio/root test and power series. The example in the book for ratio/root test is:

    Sum of: (1+n)^(n^2)
    from n=1 to infiniti

    The book says to take limit as n goes to infinity of an over the limit as n goes to infinity of an+1 . I don't get how this helps to determine if the series is convergent. Could someone walk me through this problem, and tell me how i know to use ratio or root test.

    The book simply says the answer and doesn't include any steps as to how it got that answer

    Also the power series does not make sense to me. I don't get how you identify when you use it. The example in the book is:

    3/(x^2 - x - 2)

    It says write it as a power series. It gives the answer but doesnt describe how to do that. And from this example i'm suppose to be able to do the rest of my homework, but i don't get how to write this as a power series. Could someone walk me through the steps of this one too?
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by fogel1497 View Post
    Okay so i missed the days in class when we went over power series and ratio/root test because of another exam. Reading my book i cant figure out how to do ratio/root test and power series. The example in the book for ratio/root test is:

    Sum of: (1+n)^(n^2)
    from n=1 to infiniti

    The book says to take limit as n goes to infinity of an over the limit as n goes to infinity of an+1 . I don't get how this helps to determine if the series is convergent. Could someone walk me through this problem, and tell me how i know to use ratio or root test.

    The book simply says the answer and doesn't include any steps as to how it got that answer

    Also the power series does not make sense to me. I don't get how you identify when you use it. The example in the book is:

    3/(x^2 - x - 2)

    It says write it as a power series. It gives the answer but doesnt describe how to do that. And from this example i'm suppose to be able to do the rest of my homework, but i don't get how to write this as a power series. Could someone walk me through the steps of this one too?
    The first should actually be that if \sum{a_n} is an infinite series, then the following is true: Let \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=  R. Then \sum{a_n} converges iff R<1

    For the second consider this, you should know that

    \forall{x}\backepsilon|x|<1~\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n

    So now consider that

    \frac{3}{x^2-x-2}=\frac{-3}{2}\frac{1}{1-\frac{x^2-x}{2}}

    So then we can conclude that

    \forall{x}\backepsilon\left|\frac{x^2-x}{2}\right|<1~\frac{3}{x^2-x-2}=\frac{-3}{2}\sum_{n=0}^{\infty}\left(\frac{x^2-x}{2}\right)^n
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