1 . 数列极限理论是数学中重要的理论之一,它研究的是数列中数值的变化趋势和性质.数列极限概念作为微积分的基础概念,它的产生与建立对微积分理论的创立有着重要的意义.请认真理解下述3个概念.
概念1:对无穷数列
,称
为数列
的各项和.
概念2:对一个定义域为正整数集的函数
,如果当
趋于正无穷大时,
的值无限趋近于一个常数
,即当
时,
,就说常数
是
的极限值,记为
.如:
,当
时,由反比例函数的性质可知
,即记为
.当
(
为常数)时,
.
概念3:对无穷数列
,其各项和为
,若当
时,
(
为常数),即
,则称该数列的和是收敛的,
为其各项和的极限;若当
时,其各项和
的极限不存在,则称该数列的和是发散的,其各项和的极限不存在.
试根据以上概念,解决下列问题:
(1)在无穷数列
中,
,求数列
的各项和
的极限值;
(2)在数列
中,
,讨论数列
的和是收敛的还是发散的;
(3)在数列
中,
,求证:数列
的和是发散的.
概念1:对无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88c434a9e76de70c0af36c324e1fd48d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
概念2:对一个定义域为正整数集的函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4136968179e01108272af01324034127.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6784211a2342d9d829bd95e15b549b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0057f1742dc20e867bcbc29e6475773a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40cd74412213ddb92f6b4637888cf3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5cfc53624067d3c8e01f09361295dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc76422aeaa304648c34cd1c6c0674e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f4eb29a351c1efa18e8e45d083491df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/961ea9a98e63ba37f650fde96c774026.png)
概念3:对无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b06ea1200f943f6eb160b49e584b4335.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f614310a33734a2d82f0d84c627028e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2cb2108952d47acb4f0a9518cbef443.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a107eb946e0fe41629c644b7628d5cba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b06ea1200f943f6eb160b49e584b4335.png)
试根据以上概念,解决下列问题:
(1)在无穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5beb1d3014af78f347ea9cf3661881cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccecde965d7557d5ee35dea8ae7164a3.png)
(2)在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1111d85a7c8b1842e38b5d59da90954.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f329b217e1051b23f0d61023cdc6e69.png)
(3)在数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f34f1354aaa4fa27de5215098e0b1e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ab0309e2cd35585ea9fb2cc3017abf.png)
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2 . 对于正整数
,最接近
的正整数设为
,如
,记
,从全体正整数中除去所有
,余下的正整数按从小到大的顺序排列得到数列
,则数列
的前8项和为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d249094ecb996458e35182d6b461299.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c82c7591e80b01a6fac3f7cd499514d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46752bd68b97f8cb69b26e14acdc468.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
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2023-02-03更新
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3卷引用:河南省驻马店市2022-2023学年高三上学期期末统一考试数学(理科)试题
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3 . 已知数列:
;
,
,
;
,
,…,
;…,
,
,
,…
;…,则此数列的前2036项之和为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eff998d034284391ca064755fa6bf1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5337f3688eb5261c3d1c4682eea6db8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57717fff0ecfa65745ab87175e4ef4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51dd36cdc994842315cb6b33ba6e2110.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9792c7803677e0a6a60d7b4b1d26419d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecabfaab794a6218e082092a1b057763.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/922dab5a2938593bf50a3b580d568f18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5f294032347d80dba06576e4f6c028f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48b35159c34234ffee63bc7085b84ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/682ab5018df0aed155a64ac89fd98883.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cbf3b2926a01e58de6d05d8bc949e21.png)
A.1024 | B.2048 | C.1018 | D.1022 |
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8卷引用:2019届河南省郑州外国语学校高三全真模拟(三)数学(理)试题
2019届河南省郑州外国语学校高三全真模拟(三)数学(理)试题安徽省六安市毛坦厂中学2019届高三3月月考理科数学试题【市级联考】安徽省巢湖市2019届高三年级三月份联考数学(理科)试题(已下线)专题6.4 等差、等比数列与数列求和(练)【理】-《2020年高考一轮复习讲练测》(已下线)专题19 数列求和-冲刺2020高考跳出题海之高三数学模拟试题精中选萃内蒙古赤峰二中2018-2019学年高一下学期第一次月考数学(理)试题江苏省苏州市高新区第一中学2020-2021学年高二上学期期中数学试题四川省成都市第十七中学2019-2020学年高一下学期期中数学试题
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4 . 已知函数
(其中
)的图像经过点
,令
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4025d2798141ceb5489785638e6eb36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91f8e82e985f9c7bd22d79ebaa8a2a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64a297d9b18487cde772be5c7d2676b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03193aaea57bc6f935f2c299ecc2a238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c58e99e3c3a921cd712906b0a3b1091d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4025d2798141ceb5489785638e6eb36a.png)
A.2019 | B.![]() | C.6057 | D.![]() |
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7卷引用:2020届河南省非凡联盟高三调研考试数学(理)试题
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名校
5 . 设
表示正整数
的所有因数中最大的奇数与最小的奇数的等差中项,数列
的前
项和为
,那么
的值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3505e1c60ee386b67c2ff40673ae52c.png)
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6 . “斐波那契”数列由十三世纪意大利数学家斐波那契发现.数列中的一系列数字常被人们称之为神奇数.具体数列为1,1,2,3,5,8
,即从该数列的第三项数字开始,每个数字等于前两个相邻数字之和.已知数列
为“斐波那契”数列,
为数列
的前
项和,若
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2fd03ad33bb566949faeb2fb8332a37.png)
__________ .(用M表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b5b95d2bc97a4c65a25e908bf97a34a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a3f1f8813adcedd56b471961b2ad6a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b5b95d2bc97a4c65a25e908bf97a34a.png)
![](https://img.xkw.com/dksih/QBM/2018/12/21/2101389398794240/2105684073717760/STEM/53210b5d5f5f4a24a53ff509b16ec81c.png?resizew=8)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dde757049c61d4585879edfa7d695ecd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2fd03ad33bb566949faeb2fb8332a37.png)
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