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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解题方法
2 . 设公差不为零的等差数列
的前n项和为
,且
,
.
(1)求数列
的通项公式;
(2)若
=
,数列
的前n项和为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36183db0759eec0e108274d229fcd00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f9b2392bd67dc2427bf0654ec0d7857.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e15f15639c3463b0dfd91797af369b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae6358af7332d7609bf8d18467487d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca91cc521bd5796cac29169a3ca79d5a.png)
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解题方法
3 . 已知
,
,
分别为
三个内角A,
,
的对边,
.
(1)求证:
;
(2)若
为锐角三角形,且
,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3c59168d3b6c99fe4fcf2f001e66b0b.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd813767c3263437ee6e96a1ab75bb7.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/beb4f9907b03369bbb3b7fa91c6a9d97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2024-05-08更新
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4卷引用:河南省部分学校2023-2024学年高一下学期6月联考数学试题
河南省部分学校2023-2024学年高一下学期6月联考数学试题甘肃省酒泉市四校联考2023-2024学年高一下学期5月期中考试数学试题(已下线)专题02 第六章 解三角形及其应用-期末考点大串讲(人教A版2019必修第二册)(已下线)专题05 解三角形(2)-期末考点大串讲(人教B版2019必修第四册)
4 . 已知数列
的首项
且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27abc0ddb1696ba33720fe25cb57fb75.png)
(1)证明:
是等比数列;
(2)求数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8268477a9da05189133a427668aa4c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27abc0ddb1696ba33720fe25cb57fb75.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5816ff5a1585c9b748b13f5b8bd0b6f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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5 . 如图,在
中,
.
为等边三角形.
(2)试问当
为何值时,
取得最小值?并求出最小值.
(3)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98f6cab08aec875683b343184a701c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)试问当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a116fa5c4c6d67aa3454ef87826718.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50eae187c6532ec0abf6fd40b0cc2da6.png)
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3卷引用:河南省开封市多校2023-2024学年高一下学期第三次月考数学试题
6 . 在
中,角
的对边分别是
,且
.
(1)证明:
.
(2)若
,求
的取值范围,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d82705b3f982b0de94cc9954cefc07a.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baa8d75a6638e08eedbff8662267da6f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3320a13248a3a1208ff6ee85c9d26f36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
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7 . 已知数列
满足
,
.
(1)证明:数列
为等比数列;
(2)在
与
之间插入
个数,使得这
个数组成公差为
的等差数列,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57bfc1f8772f31748bfdc280d0712fc0.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e2de706dc5f0439b989273a5367f63a.png)
(2)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/217b927efe12a98e1082ecd7f035b921.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4792fd59c4ca11ff03dc32e367c3983f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d1ae060458f733025fc82f7c7b14f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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8 . 已知数列
和
满足
,且数列
的前n项和为
.
(1)求数列
的通项公式及前n项和
;
(2)令
,记数列
前n项和为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a004a05b0d5a466afb2f7ae954e7bd12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67896de3716021a267fd4bc8b5a8b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3860c78a8d25ac6b5c1cff5ebbd960fc.png)
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9 . 如图,在平面四边形ABCD中,已知
,
,
为等边三角形,记
,
.
,求
的面积;
(2)证明:
;
(3)若
,求
的面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4aca5534bce25acaeb7379deed8f8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/833cfda415649b832cc136caed392753.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3202b1d9f838c32ab5765ce647d96b6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9940f9f7b1e9a26ab25527406be4d712.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3416881a6f67d05fe6b67787047fc86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9276a56a6c0ed7ecbc4e6e5e19af53b2.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c98466484e09a9a4ff6b10785d6715.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
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2卷引用:河南省安阳市林州市第一中学2023-2024学年高一下学期5月月考数学试题
名校
解题方法
10 . 已知数列
的前n项和为
,且满足
.
(1)证明:数列
从第二项起是等比数列,并求数列
的通项公式;
(2)设
,求数列
的前n项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3768db0f2e2881b810d44ddc39ff295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a37a022e684866a2f330629e1c4ec87.png)
(1)证明:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c895d4ce5ce82ef9b311b9369b4de11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f03b007be99a17613246b5ea1ff86d75.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feeb72938e48999e1dcbede230aacd3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
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