两类形式幂级数的递推公式.pdf

Pure Mathematics 理论数学, 2014, 4, 130-137 Published Online July 2014 in Hans. /journal/pm /10.12677/pm.2014.44020 The Recursive Formula of Two Kinds of Formal Power Series Yanli Chen, Wanhui Ji Yinchuan Energy Institute, Yinchuan Email: chenyanli8866.hi@163.com th th th Received: Jun . 10 , 2014; revised: Jul. 8 , 2014; accepted: Jul. 16 , 2014 Copyright © 2014 by authors and Hans Publishers Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). /licenses/by/4.0/ Abstract By using the trigonometric function, the hyperbolic function and the series expansion of their product, the expansions and the recursive formula of these two forms of power series ∞ −1 n 2p +1 ∞ −1 n 4p +1 ( ) π 2p ( ) π [ ∑ 2p +1 22p +1 ( x ’s ratio of secx series) and ∑ 4p +1 4 ⋅24p +1 n=−∞ (2n +1) n 1 (2n +1) cosh (n +1 2)π ( x 4p ’s ratio of secxsechx series )] are derived. At the same time, we prove them one by one using the residue theorem. Keywords Series of Trigonometric Function, Formal Power Series, Recursive Formula, Residue Theorem 两类形式幂级数的 递推公式 陈艳丽，及万会 银川能源学院，银川 Email: chenyanli8866.hi@163.com 收稿日期：2014年6月10 日；修回日期：2014年7月8 日；录用日期：2014年7月16 日 130 两类形式幂级数的递推公式 摘 要