lim(n→∞)|[(2n+3)x^(2n+2)/(n+1)!]/[(2n+1)x^(2n)/n!]|=0
x∈(-∞,+∞)
拆项
【e^x=∑(n=0~+∞)(1/n!)x^n=1+x+x^2/2!+x^3/3!+...+x^n/n!+...】
令x^2=y
∵∑(n=1~+∞)[(2n+1)/n!]y^n
=2∑(n=1~+∞)[1/(n-1)!]y^n+∑(n=1~+∞)(1/n!)y^n
=2y∑(n=1~+∞)[1/(n-1)!]y^(n-1)+∑(n=1~+∞)(1/n!)y^n
=2ye^y+e^y-1
=2x²e^x²+e^x²-1