幂级数sum from n=1 to ∞ (x^(2n)/(2n)~k)和函数的递推公式及其应用

Recurrence Formula of Sum Function of Power Series sum from n=1 to ∞ (x^(2n)/(2n)~k)

根据收敛级数的分析性质研究了幂级数sum from n=1 to ∞ (x^(2n)/(2n)~k)(k≥2)的和函数问题,用数学归纳法证明了其和函数的递推公式,由此得出k=2,3,4,…时幂级数和函数的具体表达式,进而导出几个与之相关的非初等积分的值或近似值.

The problem of sum function of power series ^∞∑n=1 x^2n/（2n）^k（k≥2） is studied according to convergent series＇ analysis property, and the recurrence formula of its sum function with induction is deduced, and obtained the concrete expression of sum function of power series when k =2, 3, 4, …, then several nonelementary integrals value or the approximate value with it derived.