ï¼1ï¼å½ n=1 æ¶ï¼å·¦è¾¹=1/2 ï¼å³è¾¹=2/4=1/2 ï¼å æ¤å·¦è¾¹>=å³è¾¹ï¼ä¸çå¼æç«ï¼
ï¼2ï¼è®¾å½ n=k æ¶ï¼ä¸çå¼æç«ï¼å³ 1/2+1/3+.........+1/k+1/(k+1)>=2k/(k+3) ï¼
åå½ n=k+1 æ¶ï¼æ
1/2+1/3+........+1/k+1/(k+1)+1/(k+2)
>=2k/(k+3)+1/(k+2)
=[2k^2+4k+k+3]/[(k+3)(k+2)]
=(2k^2+5k+3)/[(k+3)(k+2)]
=2(k+1)/(k+4)+(2k^2+5k+3)/[(k+3)(k+2)]-2(k+1)/(k+4)
=2(k+1)/(k+4)+(k^2+k)/(k^3+9k^2+26k+24)
>2(k+1)/(k+4) ï¼
è¿è¯´æå½ n=k+1 æ¶ï¼ä¸çå¼ä¹æç«ï¼
æ ¹æ®ï¼1ï¼ï¼2ï¼å¯ç¥ï¼ä¸çå¼ 1/2+1/3+........+1/n+1/(n+1)>=2n/(n+3) 对任ææ£æ´æ° n é½æç« ã
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