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南开大学2024数学分析压轴题

  • 2025-12-02 11:47:24

8.(10分) 设 是一个收敛的正项级数，令 , 若 在 上满足利普希兹条件，即存在常数 ，使得对任意实数 ，有

证明: 级数 收敛.

解析: 取,由题目条件可得

配成

令立马推出

部分和有界从而收敛.

评注: 这道题也是纸老虎，如果跟着我们做过北京邮电大学2022年的这套卷子，看到这道题可能会自然而然想到方法思路.

 

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  1. CONNECT:[ UseTime:0.000702s ] mysql:host=127.0.0.1;port=3306;dbname=rcgz;charset=utf8mb4
  2. SHOW FULL COLUMNS FROM `fenlei` [ RunTime:0.000915s ]
  3. SELECT * FROM `fenlei` WHERE `fid` = 0 [ RunTime:0.000291s ]
  4. SELECT * FROM `fenlei` WHERE `fid` = 1 [ RunTime:0.000253s ]
  5. SELECT * FROM `fenlei` WHERE `fid` = 2 [ RunTime:0.000244s ]
  6. SELECT * FROM `fenlei` WHERE `fid` = 3 [ RunTime:0.000211s ]
  7. SELECT * FROM `fenlei` WHERE `fid` = 4 [ RunTime:0.000220s ]
  8. SELECT * FROM `fenlei` WHERE `fid` = 5 [ RunTime:0.000190s ]
  9. SELECT * FROM `fenlei` WHERE `fid` = 6 [ RunTime:0.000203s ]
  10. SELECT * FROM `fenlei` WHERE `fid` = 7 [ RunTime:0.000207s ]
  11. SELECT * FROM `fenlei` WHERE `fid` = 37 [ RunTime:0.000206s ]
  12. SHOW FULL COLUMNS FROM `article` [ RunTime:0.000623s ]
  13. SELECT * FROM `article` WHERE `id` = 182276 LIMIT 1 [ RunTime:0.000525s ]
  14. UPDATE `article` SET `lasttime` = 1764660430 WHERE `id` = 182276 [ RunTime:0.005516s ]
  15. SELECT * FROM `fenlei` WHERE `id` = 4 LIMIT 1 [ RunTime:0.000257s ]
  16. SELECT * FROM `article` WHERE `id` < 182276 ORDER BY `id` DESC LIMIT 1 [ RunTime:0.000431s ]
  17. SELECT * FROM `article` WHERE `id` > 182276 ORDER BY `id` ASC LIMIT 1 [ RunTime:0.000412s ]
  18. SELECT * FROM `article` WHERE `id` < 182276 ORDER BY `id` DESC LIMIT 10 [ RunTime:0.001026s ]
  19. SELECT * FROM `article` WHERE `id` < 182276 ORDER BY `id` DESC LIMIT 10,10 [ RunTime:0.005643s ]
  20. SELECT * FROM `article` WHERE `id` < 182276 ORDER BY `id` DESC LIMIT 20,10 [ RunTime:0.008111s ]
0.098581s