证明数列是无穷小量 证明数列为无穷小量
\u600e\u4e48\u6309\u5b9a\u4e49\u8bc1\u660e\u6570\u5217\u662f\u65e0\u7a77\u5c0f\u91cf\uff0c\u4e3e\u4f8b\u8bf4\u660e\u7b2c\u4e00\u4e2a\u597d\u5417\uff1f\u65e0\u7a77\u5c0f\u91cf\u5b9a\u4e49\uff0c\u5feb\u6765\u770b\u770b\u5427
\u540e\u9762\u662f\u4ea4\u9519\u7ea7\u6570\uff0c(-1)^n*1/n \uff0c\u7531\u4e8e 1/n \u8d8b\u4e8e 0 \uff0c\u56e0\u6b64\u7ea7\u6570\u6536\u655b\uff08\u5b9e\u9645\u4e0a\u8d8b\u4e8e ln2\uff09\uff0c
\u800c\u524d\u9762 1/n \u662f\u65e0\u7a77\u5c0f\u91cf\uff0c\u56e0\u6b64\u7ea7\u6570\u662f\u65e0\u7a77\u5c0f\u91cf \u3002
绛旓細n-1)+1/n =(1-1/2)+(1/3-1/4)+鈥+[1/(n-2)-1/(n-1)]+1/n 锛1-1/2 =1/2 1-1/2+1/3-1/4+鈥-1/(n+1)+1/n =1-(1/2-1/3)-鈥-[1/(n-1)-1/n]锛1 鈭1-1/2+1/3-1/4+鈥+(-1)^(n+1)/n渚濈劧鏄湁鐣岄噺銆傛牴鎹湁鐣岄噺脳鏃犵┓灏=鏃犵┓灏忓嵆鍙
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绛旓細楂樻暟鏃犵┓澶ф棤绌峰皬璇佹槑棰,鍦ㄧ嚎绛,鎬 (1)璇佹槑鏁板垪{(2n^3-5n+1)/(5n^2-4n-4)涓烘棤绌峰ぇ閲(2)璇佹槑鏁板垪{[n+(-1)^n]/(n^2-1)}涓烘棤绌峰皬閲(3)璇佹槑鏁板垪{(n^2+1)/(2n+1)}... (1)璇佹槑鏁板垪{(2n^3-5n+1)/(5n^2-4n-4)涓烘棤绌峰ぇ閲(2)璇佹槑鏁板垪{[n+(-1)^n]/(n^2-1)}涓烘棤绌峰皬閲(3)...
绛旓細鏁板垪楂橀樁鏃犵┓灏忔槸寰Н鍒嗕腑鐨勪竴涓噸瑕佹蹇碉紝瀹冨彲浠ョ敤鏉璇佹槑鏋侀檺鐨勫瓨鍦ㄦу拰鍞竴鎬с備互涓嬫槸浣跨敤鏁板垪楂橀樁鏃犵┓灏忔潵璇佹槑鏋侀檺鐨勪竴鑸楠わ細1.棣栧厛锛屾垜浠渶瑕佹壘鍒颁竴涓暟鍒楋紝瀹冪殑姣忎竴椤归兘鏄師鏁板垪鐨鏃犵┓灏忛噺銆傝繖涓暟鍒楄绉颁负鍘熸暟鍒楃殑楂橀樁鏃犵┓灏忔暟鍒椼2.鐒跺悗锛屾垜浠渶瑕佽瘉鏄庤繖涓珮闃舵棤绌峰皬鏁板垪鐨勬瀬闄愬瓨鍦ㄤ笖绛変簬鍘...
绛旓細0<3^n/n!=(3/1)*(3/2)*(3/3)*(3/4)*...*(3/(n-1))*3/n<(3/1)*(3/2)*(3/3)*(4/4)*(5/5)*(6/6)*...*(n-1)/(n-1)*3/n=(9/2)*(3/n)鎵浠 0<3^n/n锛<(9/2)*(3/n)褰搉瓒嬭繎浜庢棤绌峰ぇ鏃讹紝锛9/2)*(3/n)涓烘棤绌峰皬閲 0<3^n/n锛<(9/2)*...
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绛旓細鍥炵瓟锛歯!<2^n
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