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Interval of convergence

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  • 23-10-2011 4:32pm
    #1
    irishdude11
    Registered Users Posts: 377 ✭✭


    I have a series that is the sum from 0 to infinity of -
    (-1)^n * x^n

    And I am looking for the interval of convergence.

    If the (-1)^n wasn't there I could say this is a geometric series were r = x and then that would be convergent where |r| = |x| < 1.
    Then it would have an interval of convergence (-1, 1).

    However the (-1)^n has thrown me...I dont think I can say this is a geometric series now so how can I get its interval of convergence?


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    #2
    ZorbaTehZ
    Registered Users Posts: 2,149 ✭✭✭ZorbaTehZ


    You can use a comparison with the geometric series or more simply, you could use Hadamard's formula.


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    #3
    irishdude11
    Registered Users Posts: 377 ✭✭irishdude11


    We havent covered Hadamards formula so I cant use that.

    I dont really get how I can compare it with a geometric series...what about the (-1)^n?


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    #4
    irishdude11
    Registered Users Posts: 377 ✭✭irishdude11


    Im thinking if I use the alternating series test then the series only converges when x^n approaches 0 as n approaches infinity. So therefore it only converges when |x| < 1 so the interval of convergence is (-1,1). Does that look right?


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    #5
    MathsManiac
    Registered Users Posts: 1,595 ✭✭✭MathsManiac


    It's geometric anyway, with common ratio (-x).


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