如题所述
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ï¼1ï¼è§å¯ï¼åç°è§å¾ï¼åæ¯æ¯è¿ç»ä¸ä¸ªæ´æ°ä¹ç§¯ï¼ç¹æ®é®é¢ä¸è¬åæ¯ä»£æ°æ¢ç´¢ï¼ç¬¬n个å¼å为ï¼1/(n(n+1)(n+2))=(1/2)[(1/n)-(1/(n+1))+(1/(n+2)-(1/(n+1))]说æï¼æ¥èªæ´å¼åå¼äºåçå¯åï¼
1/(n(n+1))=(1/n)-(1/(n+1)
ä»èæ¯ä¸é¡¹é½å¯ä»¥ï¼å为(1/2)[(1/1)-(1/(1+1))+(1/(1+2)-(1/(1+1))]=(1/2)[(1-(1/2)+(1/3ï¼-(1/2)]ï¼
(1/2)[(1/2)-(1/3))+(1/4)-(1/3)]ï¼
(1/2)[(1/3)-(1/4))+(1/5)-(1/4)]ï¼
(1/2)[(1/4)-(1/5))+(1/6)-(1/5)]ï¼
(1/2)[(1/5)-(1/6))+(1/7)-(1/6)];
ä¸è¿°å个å¼åå å ï¼å¾å°ï¼æµæ¶ä¸é´å©ä¸å2个æ°åæå两个æ°ï¼
(1/2)[(1-(1/2))+(1/7)-(1/6)]=5/21
ï¼2ï¼åæåä¸ï¼æåçå¼åè¥ä¸º1/ï¼1+2+3+4+â¦â¦+n)
äºå®ä¸åæ¯ç¨å ¬å¼ï¼1+2+3+4+â¦â¦+n=n(n+1)/2
å¾å°æåçå¼å为ï¼2/(n(n+1))=2[(1/n)-(1/(n+1))]
äºæ¯ï¼åæ¥çå¼åå¯ä»¥å形为ï¼2[(1/1)-(1/2)]+2[(1/2)-(1/3)]+â¦â¦+2[(1/10)-(1/11)]=2[1-1/11]
=20/11
å³çæ¡æ¯ï¼ï¼1ï¼5/21ï¼2ï¼20/11
ï¼1ï¼è§å¯ï¼åç°è§å¾ï¼åæ¯æ¯è¿ç»ä¸ä¸ªæ´æ°ä¹ç§¯ï¼ç¹æ®é®é¢ä¸è¬åæ¯ä»£æ°æ¢ç´¢ï¼ç¬¬n个å¼å为ï¼1/(n(n+1)(n+2))=(1/2)[(1/n)-(1/(n+1))+(1/(n+2)-(1/(n+1))]说æï¼æ¥èªæ´å¼åå¼äºåçå¯åï¼
1/(n(n+1))=(1/n)-(1/(n+1)
ä»èæ¯ä¸é¡¹é½å¯ä»¥ï¼å为(1/2)[(1/1)-(1/(1+1))+(1/(1+2)-(1/(1+1))]=(1/2)[(1-(1/2)+(1/3ï¼-(1/2)]ï¼
(1/2)[(1/2)-(1/3))+(1/4)-(1/3)]ï¼
(1/2)[(1/3)-(1/4))+(1/5)-(1/4)]ï¼
(1/2)[(1/4)-(1/5))+(1/6)-(1/5)]ï¼
(1/2)[(1/5)-(1/6))+(1/7)-(1/6)];
ä¸è¿°å个å¼åå å ï¼å¾å°ï¼æµæ¶ä¸é´å©ä¸å2个æ°åæå两个æ°ï¼
(1/2)[(1-(1/2))+(1/7)-(1/6)]=5/21
ï¼2ï¼åæåä¸ï¼æåçå¼åè¥ä¸º1/ï¼1+2+3+4+â¦â¦+n)
äºå®ä¸åæ¯ç¨å ¬å¼ï¼1+2+3+4+â¦â¦+n=n(n+1)/2
å¾å°æåçå¼å为ï¼2/(n(n+1))=2[(1/n)-(1/(n+1))]
äºæ¯ï¼åæ¥çå¼åå¯ä»¥å形为ï¼2[(1/1)-(1/2)]+2[(1/2)-(1/3)]+â¦â¦+2[(1/10)-(1/11)]=2[1-1/11]
=20/11
å³çæ¡æ¯ï¼ï¼1ï¼5/21ï¼2ï¼20/11
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