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Math Help - Question about the Uniqueness Theorem of Power Series

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    Question about the Uniqueness Theorem of Power Series

    Let f(T)= \sum a_nT^n and g(T)= \sum b_nT^n be two convergent power series. Suppose that f(x)=g(x) for all x in an infinite set having 0 as a point of accumulation. Then f(T)=g(T).

    Proof.

    Define h=f-g= \sum (a_n=b_n)T^n, then h(x)=0 for an infinite set x having 0 as a point of accumulation. Now, I also know that if h(0)=0, then there exist a neighborhood centered at 0 such that h is non-constant (I actually understand the proof of that).

    But how do I incorporate this idea into the proof to arrive at the conclusion that h is 0 everywhere? Thank you.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let f(T)= \sum a_nT^n and g(T)= \sum b_nT^n be two convergent power series. Suppose that f(x)=g(x) for all x in an infinite set having 0 as a point of accumulation. Then f(T)=g(T).

    Proof.

    Define h=f-g= \sum (a_n-b_n)T^n, then h(x)=0 for an infinite set x having 0 as a point of accumulation. Now, I also know that if h(0){\color{red}\neq} 0, then there exist a neighborhood centered at 0 such that h is non-constant (I actually understand the proof of that).

    But how do I incorporate this idea into the proof to arrive at the conclusion that h is 0 everywhere? Thank you.
    (note the inequality in red; I guess this is what you meant?)

    Let c_n=a_n-b_n so that h(x)=\sum_{n=0}^\infty c_n x^n.

    Suppose by contradiction that h is not 0 everywhere. Then there is a least index n_0\geq 0 such that c_{n_0}\neq 0, and we can factor the sum to write h(x)=x^{n_0}\sum_{n=0}^\infty c_{n_0+n} x^n=x^{n_0} \left(c_{n_0}+c_{n_0+1}x+\cdots\right), so that you can see we have

    c_{n_0}=\lim_{x\to0^+}\frac{h(x)}{x^{n_0}}

    This limit is also the limit along any sequence (x_n)_n that converges to 0. However, you know there is such a sequence where h is zero. We deduce from the above limit that c_{n_0}=0, a contradiction.
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