10.数学归纳法,数列
极限
第10讲数学归纳法、数列极限
一、知识要点
1.数学归纳法及其证明步骤
2.数列极限
3.数列极限的四则运算性质
4.无穷数列的各项和
二、经典例题
1.数学归纳法
例1.用数学归纳法证明:
(1)«Skip Record If...»
(2)设«Skip Record If...»,证明对一切«Skip Record If...»的自然数,等式
«Skip Record If...»均成立
例2.«Skip Record If...»,用数学归纳法证明:
(1)«Skip Record If...»能被13整除
(2)«Skip Record If...»能被9整除
例3.(1)数列«Skip Record If...»满足«Skip Record If...»,猜想并证明«Skip Record If...»的一个通项公式
(2)数列«Skip Record If...»的前«Skip Record If...»项和为«Skip Record If...»,
当«Skip Record If...»时,«Skip Record If...»,求«Skip Record If...»,并求证
«Skip Record If...»是等比数列
2.数列的极限
例4.求下列各个数列极限
(1)«Skip Record If...» (2)«Skip Record If...»
(3)«Skip Record If...»
例5.求下列各个数列极限
(1)«Skip Record If...» (2)«Skip Record If...»
例6.求下列各个数列极限:
(1)«Skip Record If...» (2)«Skip Record If...»
例7.计算:(1)«Skip Record If...»
(2)«Skip Record If...»
(3)«Skip Record If...»
(4)«Skip Record If...»
3.无穷等比数列各项和以及应用
例8.利用等比数列各项和,计算:
(1)«Skip Record If...»
(2)«Skip Record If...»
(3)«Skip Record If...»
«Skip Record If...»
例9.已知函数«Skip Record If...»,方程«Skip Record If...»的根«Skip Record If...»,称为函数«Skip Record If...»的不动点;若«Skip Record If...»,«Skip Record
If...»«Skip Record If...»,则称«Skip Record If...»为由函数«Skip Record If...»导
出的数列设函数
«Skip Record If...»
(1)求函数«Skip Record If...»的不动点«Skip Record If...»
(2)设«Skip Record If...»,«Skip Record If...»是由函数«Skip Record If...»导出的数列,对(1)中的两个不动点«Skip Record If...»(«Skip Record
If...»),求证:数列«Skip Record If...»是等比数列,并求«Skip
Record If...»
例10.已知«Skip Record If...»满足«Skip Record If...»,且«Skip Record If...»,是否存在实数«Skip Record If...»,使得«Skip Record If...»对于任何«Skip Record If...»都成立,证明你的结论。