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Math Help - Abel's Limit Theorem

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    Abel's Limit Theorem

    Theorem. If  \sum_{0}^{n} a_{n} converges then  f(z) = \sum_{0}^{\infty} a_{n}z^{n} tends to  f(1) as  z approaches  1 in such a way that  |1-z|/(1-|z|) remains bounded.

    Here was assuming that R=1 and convergence takes place at  z = 1 .


    But we let  R = k and  z = k for example, right?
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    Quote Originally Posted by manjohn12 View Post
    Theorem. If  \sum_{0}^{n} a_{n} converges then  f(z) = \sum_{0}^{\infty} a_{n}z^{n} tends to  f(1) as  z approaches  1 in such a way that  |1-z|/(1-|z|) remains bounded.

    Here was assuming that R=1 and convergence takes place at  z = 1 .

    But we let  R = k and  z = k for example, right?
    That is correct. The usual statement of Abel's theorem is that if the series  f(z) = \sum_{0}^{\infty} a_{n}z^{n} has radius of convergence R, and converges at a point z_0 on the circle of convergence (so |z_0|=R), then \lim f(z) = f(z_0) as z_0\to z nontangentially (meaning that that the angle between z_0-z and the tangent at z_0 is bounded away from 0).
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