求和：Sn=1-3x+5x^2-7x^3+.+(2n+1)(-x)^n(n属于N*)

来源：学生作业帮编辑：百度作业网作业帮分类：数学作业时间：2024/04/29 08:36:58

Sn 是等差数列an = 2n-1,等比数列 bn = (-x)^(n-1)
前n+1的和
Sn -(-x) Sn = (a1b1 + a2b2+...an+1 bn+1) - (a1b2 + a2b3+ ...+ an+1 bn+2)
= a1b1 + [ (a2-a1)b2 + (a3-a2)b3 +...+ (an+1 - an)bn+1] - an+1 bn+2
= 1 + 2*[ b2+b3 +...+bn+1] - (2n+1)bn+2
= 1 + 2*[ b2+b3 +...+bn+1] - (2n+1)(-x)^(n+1)
= 1 + 2* [(b2-bn+2)/(1-(-x))] - (2n+1)(-x)^(n+1)
= 1 + 2*[ (-x - (-x)^(n+1))/(1+x) ] - (2n+1)(-x)^(n+1)
= 1 + 2(-x)/(1+x) - 2(-x)^(n+1)/(1+x) - (2n+1)(-x)^(n+1)
Sn = (Sn-(-x)Sn)/(1+x)
= 1/(1+x) -2x/(1+x)^2 - 2(-x)^(n+1)/(1+x)^2 - (2n+1)(-x)^(n+1)/(1+x)

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