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Math Help - Power series

  1. #1
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    Power series

    a_nx^n, b_nx^n are two power series with radius of convergence R1, R2, respectively.
    If R1 is different than R2, how can i prove that the radius of convergence of the power series (a_n + b_n)x^n is min{R1, R2}?
    And what can be said about the radius of cenvergence if R1=R2?

    Thanks for any help
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by rebecca View Post
    a_nx^n, b_nx^n are two power series with radius of convergence R1, R2, respectively.
    If R1 is different than R2, how can i prove that the radius of convergence of the power series (a_n + b_n)x^n is min{R1, R2}?
    And what can be said about the radius of cenvergence if R1=R2?

    Thanks for any help
    You cannot prove this because it is not true: The radius of convergence of the series \sum_n (a_n+b_n)x^n might be larger than \min\{R_1,R_2\}, but it cannot be strictly smaller.

    For a trivial example of this just take a_n := 1 and  b_n := -1 for all n. The two series individually have a radius of convergence equal to 1 but the power series with coefficients a_n+b_n=0 converges for all x\in \mathbb{R} whatever.

    The radius of convergence cannot be strictly smaller, because if |x|<\min\{R_1,R_2\}, then the two series \sum_n a_n x^n,\sum_n b_n x^n both converge absolutely and therefore their sum \sum_n a_n x^n + \sum_n b_n x^n=\sum_n (a_n +b_n)x^n also converges absoluely.
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