級數
定義
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![絕對收斂](/img/5/814/wZwpmL4MDN2cTNyAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLxIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![絕對收斂](/img/5/814/wZwpmL4MDN2cTNyAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLxIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
如果級數 各項的絕對值所構成的級數 收斂,則稱級數 絕對收斂,級數 稱為絕對收斂級數。
定理
定理1:絕對收斂級數一定收斂。
![絕對收斂](/img/5/814/wZwpmL4MDN2cTNyAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLxIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
定理2:設級數 絕對收斂,且其和等於S,則任意重排後所得的級數也絕對收斂,且有相同的和數。
注意:由條件收斂級數重排後所得的新級數,即使收斂,也不一定收斂於原來的和數。而且,條件收斂級數適當排列後,可得到發散級數,或收斂於事先任意指定的數。
定理3:若級數:
![絕對收斂](/img/9/ed7/wZwpmL2gDO0gTOzITOxMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL4QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/7/562/wZwpmLwgDNyMzM3gjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL4YzL0gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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都絕對收斂,則對所有乘積 按任意排列所得的級數 也絕對收斂,且其和等於AB。
判別方法
![絕對收斂](/img/5/814/wZwpmL4MDN2cTNyAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLxIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![絕對收斂](/img/b/693/wZwpmL2ITO0QDM4EDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL0QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
由定義可知,要知道 是否絕對收斂,只需要看 是否收斂。下面將介紹5種判別級數是否收斂的方法。
(1)【阿貝爾判別法】
![絕對收斂](/img/9/c81/wZwpmL4cDO1ADNxcDOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
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若 為單調有界數列,且級數 收斂,則級數 收斂。
(2)【狄利克雷判別法】
![絕對收斂](/img/9/c81/wZwpmL4cDO1ADNxcDOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3gzLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
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![絕對收斂](/img/3/8ad/wZwpmLzgDMxATNxMTOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/c/792/wZwpmLzcTN5czN4EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczLwEzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若數列 單調遞減,且 ,又級數 的部分和數列有界,則級數 收斂。
(3)【比式判別法】
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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設 為正項級數,且存在某正整數 及常數q 。若對一切 ,不等式 成立,則級數 收斂;若對一切 ,不等式 成立,則級數 發散。
【推論】
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/2/25b/wZwpmLzEzN1cDO1UDOwMzM1UTM1QDN5MjM5ADMwAjMwUzL1gzL2UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/3/9ee/wZwpmL1AzN5cjN2EzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/0/229/wZwpmLxcjNxkTM5gTNxMzM1UTM1QDN5MjM5ADMwAjMwUzL4UzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/1/226/wZwpmLwUTM1ATO5UzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL1czL4MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/c/54d/wZwpmL4IjM2IDM4ITOwMzM1UTM1QDN5MjM5ADMwAjMwUzLykzLzczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
設 為正項級數,且 ,則若 時,級數 收斂;若 或 時,級數 發散;若 ,則無法判斷。
(4)【根式判別法】
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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設 為正項級數,且存在某正整數 及正常數 。若對一切 ,不等式 成立,則級數 收斂;若對一切 ,不等式 成立,則級數 發散。
【推論】
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/9/61b/wZwpmLzYDO1YDNygjMxMzM1UTM1QDN5MjM5ADMwAjMwUzL4IzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![絕對收斂](/img/a/94c/wZwpmL3cDO0MTO5MTOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzL0YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/6/c42/wZwpmL3EDNykjMxAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLzgzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
設 為正項級數,且 ,則若 時,級數 收斂;若 時,級數 發散;若 ,則無法判斷。
(5)【比較原則】
![絕對收斂](/img/9/fb0/wZwpmLyADN2IjN2cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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設 和 是兩個正項級數,如果存在某個正數N,對一切n>N,都有: ,若級數 收斂,則,級數 也收斂;若級數 發散,則, 也發散。
無窮積分
定義
![絕對收斂](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/e/a8c/wZwpmLxMjN3kDO3QjN0kTO0UTMyITNykTO0EDMwAjMwUzL0YzL3IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/f/cae/wZwpmL2QzN4AjNxYzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL2czLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![絕對收斂](/img/a/959/wZwpmLzgTMzYzM5kzNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5czL3gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
1. 若函式 在任何有限區間 上可積,且無窮積分 收斂,則稱 絕對收斂。
![絕對收斂](/img/3/78e/wZwpmLyMDNzQDM5EjNxADN0UTMyITNykTO0EDMwAjMwUzLxYzL2AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/f/dfb/wZwpmLzUjM3EjNzQTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![絕對收斂](/img/f/cae/wZwpmL2QzN4AjNxYzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL2czLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![絕對收斂](/img/d/93b/wZwpmLzYTNyYzM5kzNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5czL3IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
2.函式在區間上連續,且無窮限積分收斂,則稱絕對收斂
定理
![絕對收斂](/img/a/959/wZwpmLzgTMzYzM5kzNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5czL3gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![絕對收斂](/img/a/325/wZwpmL2czMzMTO3QjN0kTO0UTMyITNykTO0EDMwAjMwUzL0YzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![絕對收斂](/img/d/fa8/wZwpmL0YDNzgDO4MTOwMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLxIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/2/df1/wZwpmL0EDO3gjM3ETOwMzM1UTM1QDN5MjM5ADMwAjMwUzLxkzL4UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
1. 無窮積分 收斂的充要條件是:任給 ,存在 ,只要 ,便有:
![絕對收斂](/img/d/9dd/wZwpmLyEzM2cDNwgTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL4kzL1QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/f/cae/wZwpmL2QzN4AjNxYzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL2czLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![絕對收斂](/img/e/e9a/wZwpmL2YTO2QTNxQTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLyMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
2.收斂的充要條件是:存在上界
判定方法
(1)【比較判別法】
![絕對收斂](/img/9/0fc/wZwpmL4YTMzEjMyQTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLzgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/c/2a8/wZwpmLzIDM3cjNwcTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3kzLwczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設定義在上的兩個函式f 和 g 都在任意有限區間上可積,且滿足
![絕對收斂](/img/1/773/wZwpmL2czMxcTM3cTOwMzM1UTM1QDN5MjM5ADMwAjMwUzL3kzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![絕對收斂](/img/4/07e/wZwpmLxgzNwEDMzgzMxMzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLzEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![絕對收斂](/img/f/cae/wZwpmL2QzN4AjNxYzNxMzM1UTM1QDN5MjM5ADMwAjMwUzL2czLxYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
則當收斂時,必定收斂。
(2)【狄利克雷判別法】
![絕對收斂](/img/4/73b/wZwpmLwEDO3gjN4YzMxMzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL3YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/9/0fc/wZwpmL4YTMzEjMyQTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLzgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/8/36b/wZwpmLycTN3YDNxkjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL4AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![絕對收斂](/img/9/0fc/wZwpmL4YTMzEjMyQTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLzgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/d/04d/wZwpmLxQDNwUTNzAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczL1QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![絕對收斂](/img/3/9d8/wZwpmLyIDM5AzNxMTOxMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLyYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若在上有界,在上當 時單調趨於0,則收斂。
(3)【阿貝爾判別法】
![絕對收斂](/img/a/959/wZwpmLzgTMzYzM5kzNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5czL3gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![絕對收斂](/img/8/36b/wZwpmLycTN3YDNxkjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL4AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![絕對收斂](/img/9/0fc/wZwpmL4YTMzEjMyQTOxMzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLzgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![絕對收斂](/img/3/9d8/wZwpmLyIDM5AzNxMTOxMzM1UTM1QDN5MjM5ADMwAjMwUzLzkzLyYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若收斂,在上單調有界,則收斂。
無論無窮級數還是無窮積分,它們都是要么發散,要么條件收斂,要么絕對收斂,三者必居其一。