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A New Theory of Infinite Series and its Applications in Training Weight Function Neural Networks

A new theory of infinite series is proposed in this paper, some new important theorems for function expansion and infinite series are also proposed. Unlike Taylors expansion, the expansion generated by a function is not the form of polynomials. In general, the performance of convergence is much better than that obtained by Taylor's Series. The new important theorems lay the foundation for the new theory of infinite series and applications. To describe the performance of the new results obtained in this paper, an example given in this paper shows that the region of convergence is much larger than that of Taylors series. The new infinite series can keep some important properties of original functions. Weight function neural networks are also used to training feedforward neural networks based on the new theory proposed in this paper.