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Thread: radius of convergence

  1. #1
    Feb 2009

    radius of convergence

    how come when u multiply a series by x^n or when u multiply a series by a constant, it doesnt change the radius of convergence? ex: the power series representation for e^x, the ROC doesnt change for the series represenation of x^2e^x, or if a constant was multiplied to the series represenation for e^x, i know that if u use the ratio test it's true that the ROC doesnt change, but like can someone explain to me like why intuitively it would be true?
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  2. #2
    MHF Contributor matheagle's Avatar
    Feb 2009
    What we're doing doing is a limit comparison to the geometric series.
    That's why we compare the next term to our present term.

    And since $\displaystyle \biggl({x+1\over x}\biggr)^2\to 1$ as $\displaystyle x\to\infty$

    theterms of the form $\displaystyle x^2$ can't change the interior of the interval of convergence.
    However, it can possibly alter convergence/divergence at the endpoints.
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