论文信息 - On the Zeroes of the Nth Partial Sum of the Exponential Series

On the Zeroes of the Nth Partial Sum of the Exponential Series

2. GENERAL PROPERTIES. The fundamental theorem of algebra states that any polynomial of degree N (> 0) has exactly N zeroes, provided that zeroes are counted according to multiplicity. Thus, each polynomial PN (z) has exactly N zeroes. Proposition 1. For each N ≥ 1 all zeroes of PN (z) are simple. Proof. If r were a multiple root of PN (z), then r would also be a zero of P ′ N (z) = PN−1(z). But then 0 = PN (r) − PN−1(r) = r N N ! , implying that 0 is a zero of PN (z), an obvious contradiction.

Stephen M. Zemyan | S. M. Zemyan

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