On the range of $\sum_{n=1}^\infty\pm c_n$

Let $\{c_n\}_{n=1}^\infty$ be a sequence of complex numbers. In this paper we answer when the range of $\sum_{n=1}^\infty\pm c_n$ is dense or equal to the complex plane. Some examples are given to explain our results. As its application, we calculate the Hausdorff dimension of the level sets of a Rademacher series with complex coefficients.

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