基本概念
數列
![數列極限](/img/8/778/wZwpmL2MzN5UzM2MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![數列極限](/img/3/23d/wZwpmLwADO5AjNyIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL4MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
定義 若函式的定義域為全體正整數集合,則稱
![數列極限](/img/5/448/wZwpmL2IjMwUzMxUTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1EzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數列極限](/img/3/23d/wZwpmLwADO5AjNyIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL4MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/5/beb/wZwpmL3cjNycTN0ITMzEzM1UTM1QDN5MjM5ADMwAjMwUzLyEzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
為數列。因正整數集的元素可按由小到大的順序排列,故數列也可寫作
![數列極限](/img/1/820/wZwpmL3UzMzYjN3EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL4IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/578/wZwpmLycDNwEjNykzN5ADN0UTMyITNykTO0EDMwAjMwUzL5czL1MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
或可簡單地記為,其中稱為該數列的通項。
數列極限
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/e/bc6/wZwpmL3UDO1IzM5IjN5ADN0UTMyITNykTO0EDMwAjMwUzLyYzLzgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![數列極限](/img/2/3a3/wZwpmLxUzMzgDN0cTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLzUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
定義設為數列,a為定數。若對任給的正數,總存在正整數N,使得當時有
![數列極限](/img/c/4b9/wZwpmL0MzM1EDMwIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLxMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
則稱數列收斂於a,定數a稱為數列的極限,並記作
![數列極限](/img/5/768/wZwpmL4MzMzMjN1IzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyMzLyQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
若數列沒有極限,則稱不收斂,或稱發散。
![數列極限](/img/a/325/wZwpmL2czMzMTO3QjN0kTO0UTMyITNykTO0EDMwAjMwUzL0YzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
等價定義任給,若在(a-ε,a+ε)之外數列中的項至多只有有限個,則稱數列收斂於極限a。
幾何意義
當n>N時,所有的點xn都落在(a-ε,a+ε)內,只有有限個(至多只有N個)在其外,如右圖1
![圖1](/img/b/27a/wZwpmL4gTMxAzNyEjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLzEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
性質
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
唯一性 若數列 收斂,則它只有一個極限。
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/0/a26/wZwpmL1MDO4YjM5cTO4kzM0UTMyITNykTO0EDMwAjMwUzL3kzL3IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
有界性 若數列 收斂,則 為有界數列,即存在正數 ,使得對一切正整數n有
![數列極限](/img/9/385/wZwpmL3QTM2YDM0IjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLzAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![數列極限](/img/d/8ef/wZwpmLyYjMwEDM1cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL4IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/2/024/wZwpmL4QzMxkzM1EDNxMDN0UTMyITNykTO0EDMwAjMwUzLxQzL0IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![數列極限](/img/a/4bb/wZwpmLzczMygjN3UjM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL1EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/a/135/wZwpmLxQTOwMTN5czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL1gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/2/3a3/wZwpmLxUzMzgDN0cTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLzUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/9/bdd/wZwpmLzADN4kzM2gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLwQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![數列極限](/img/7/d83/wZwpmLxQTNzEDN3EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLwQzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
保號性 若 (或 ),則對 (或 ),存在正數N,使得當 時,有 (或 )。
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/e30/wZwpmLxYzNxcTO5MjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzIzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![數列極限](/img/6/5fc/wZwpmLzEjM2MzNykTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/26a/wZwpmLxMDMwcjMxUTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1EzL4IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/4/78c/wZwpmLzAjNzcjN1QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLyYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
保不等式性 設 與 均為收斂數列。若存在正數 ,使得當 時有 ,則
![數列極限](/img/e/1ac/wZwpmL0ITM2cTO1gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzL2YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/e30/wZwpmLxYzNxcTO5MjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzIzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![數列極限](/img/e/541/wZwpmLzMzMwITO0kzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL5MzL1AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
迫斂性 設收斂數列 , 都以a為極限,數列 滿足:
![數列極限](/img/6/5fc/wZwpmLzEjM2MzNykTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/26a/wZwpmLxMDMwcjMxUTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1EzL4IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/e/f16/wZwpmLzcDMzQjM5ADN2EzM1UTM1QDN5MjM5ADMwAjMwUzLwQzL3QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/e/541/wZwpmLzMzMwITO0kzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL5MzL1AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
存在正數 ,當 時有 則數列 收斂,且
![數列極限](/img/6/d9b/wZwpmL1UDMzgTMwIjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzL0MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
四則運算法則
![數列極限](/img/6/fdb/wZwpmL2UTM0IzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/e30/wZwpmLxYzNxcTO5MjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzIzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![數列極限](/img/b/82e/wZwpmLzYTNyUDM2QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL4IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/f/77f/wZwpmL4cDO2kjNyUzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLyQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/e/4aa/wZwpmL2czM1YzN2IjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
若 與 為收斂數列,則 , , 也都是收斂數列,且有
![數列極限](/img/2/68d/wZwpmLyADN2UTO1UjM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1IzLxQzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![數列極限](/img/4/cea/wZwpmL2YzM2kTOxcTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLwIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/7/238/wZwpmL0MDOxYDN3YzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLyIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![數列極限](/img/4/fda/wZwpmLwIDN5ATO5UzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLxgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/6/1a5/wZwpmLxETOycTOxEDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLxQzLyEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
若再假設 及 ,則 也是收斂數列,且有
![數列極限](/img/b/3d4/wZwpmLzEzM2EjNxQDN2EzM1UTM1QDN5MjM5ADMwAjMwUzL0QzLyIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
存在的條件
單調有界定理 在實數系中,單調有界數列必有極限。
緻密性定理 任何有界數列必有收斂的子列。
套用
![數列極限](/img/6/b6c/wZwpmLzUTN5czM2QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL4EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(1)求極限
解:
![數列極限](/img/4/20e/wZwpmL4YDN4QzM2EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLzYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/1/04e/wZwpmLxMDMwITM5EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL2IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(2)求極限
解:
因為
![數列極限](/img/f/551/wZwpmL0YDMwQzM1MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLxYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![數列極限](/img/d/17e/wZwpmLyEzN2MDOzQTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
且
![數列極限](/img/d/fb0/wZwpmL2gzN1cDNykTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL5EzLzQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![數列極限](/img/1/d6c/wZwpmL2MjMwUzM1EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLzYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
所以,由迫斂性可得
![數列極限](/img/9/03f/wZwpmLwgTO2UjNxIzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyMzL3UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)