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Math Help - absolute convergence

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    absolute convergence

    Suppose that a_k >= 0 and a_k^(1/k) → a as k → ∞. Prove that ∑ (k=1 to ∞) a_kx^k converges absolutely for all abs (x) < 1/a if a ≠ 0 and for all x in R if a=0.

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  2. #2
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    Quote Originally Posted by taypez View Post
    Suppose that a_k >= 0 and a_k^(1/k) → a as k → ∞. Prove that ∑ (k=1 to ∞) a_kx^k converges absolutely for all abs (x) < 1/a if a ≠ 0 and for all x in R if a=0.
    This is an odd question I think.
    It is nothing more than a restatement of the root test.
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  3. #3
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    Quote Originally Posted by taypez View Post
    Suppose that a_k >= 0 and a_k^(1/k) → a as k → ∞. Prove that ∑ (k=1 to ∞) a_kx^k converges absolutely for all abs (x) < 1/a if a ≠ 0 and for all x in R if a=0.

    Thanks.
    If |x| < \frac{1}{a} (assuming a\not =0 so a>0) then \limsup |a_kx^k|^{1/k} = \limsup |a_k| \cdot |x| = \frac{1}{a}|x| < 1 so it converges absolutely.
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