若数列极限唯一,那它收敛吗 数列收敛和数列极限唯一是一回事吗
\u5982\u679c\u6570\u5217\u6536\u655b\uff0c\u90a3\u4e48\u5b83\u7684\u6781\u9650\u552f\u4e00 \uff0c\u8981\u600e\u4e48\u8bc1\u660e\u8fd9\u4e2a\u5b9a\u7406\u4e00\u822c\u6559\u6750\u90fd\u6709\u8bc1\u660e\u7684\uff0c\u7ffb\u4e66\u53bb\u3002
\u6570\u5217\u6536\u655b\uff0c\u90a3\u4e48\u6570\u5217\u81ea\u7136\u6709\u6781\u9650\uff1b\u53cd\u4e4b\u4e0d\u4e00\u5b9a\uff1b
\u8bbe\u6570\u5217{a(n)}\u6536\u655b\u4e8ea\uff0ca\u662f\u6709\u9650\u6570\uff0c\u90a3\u4e48\u79f0\u6570\u5217\u6709\u6781\u9650\uff0c\u4e14\u6781\u9650\u4e3aa\uff1b
\u4f46\u6570\u5217\u6709\u6781\u9650\uff0c\u672a\u5fc5\u6536\u655b\uff0c\u5982a(n)=n\uff0ca(n)\u6781\u9650\u4e3a+\u221e\uff0c\u4f46\u4e0d\u6536\u655b\u3002
\u5047\u5982\u6570\u5217{a(n)}\u53c8\u6536\u655b\u4e8eb\uff0c\u90a3\u4e48\u4e00\u5b9a\u6709a=b\uff0c\u7528\u53cd\u8bc1\u6cd5\u8bc1\u660e\uff1a
\u6839\u636e\u6781\u9650\u7684\u5b9a\u4e49\uff0c\u82e5a\u2260b\uff0c\u4e0d\u59a8\u8bbea<b\uff1b
\u53d6e=(b-a)/2\uff0c\u5b58\u5728N1\uff0c\u4f7f\u5f97\u5f53n>N1\uff0c\u6709 a-e<a(n)<a+e\uff1b
\u53c8\u5b58\u5728N2\uff0c\u4f7f\u5f97\u5f53n>N2\uff0c\u6709 b-e<a(n)<b+e\uff1b
\u53d6N=max{N1,N2}\uff0c\u5f53n>N\u65f6\uff0c\u6709
a(n)b-e\uff0c\u90a3\u4e48 b-e<a+e\uff0c\u7531\u4e8ee=(b-a)/2\uff0c\u77db\u76fe\u3002
\u6240\u4ee5 a=b \uff0c\u5373\u5982\u679c\u6570\u5217\u6536\u655b\uff0c\u6781\u9650\u5b58\u5728\uff0c\u4e14\u6781\u9650\u552f\u4e00\u3002
绛旓細鏁板垪鏈鏋侀檺灏卞彨鏀舵暃锛骞朵笖鏋侀檺鏄鍞竴鐨勩
绛旓細鏄殑銆傝鏁板垪{Xn}锛屽鏋滃瓨鍦ㄥ父鏁癮锛屽浜庝换鎰忕粰瀹氱殑姝ｆ暟q锛堟棤璁哄灏忥級锛屾诲瓨鍦ㄦ鏁存暟N锛屼娇寰梟>N鏃讹紝鎭掓湁|Xn-a|<q鎴愮珛锛屽氨绉版暟鍒梴Xn}鏀舵暃浜巃锛堟瀬闄愪负a锛夛紝鍗虫暟鍒梴Xn}涓烘敹鏁涙暟鍒楋紙Convergent Sequences锛夈傛暟鍒楁敹鏁<=>鏁板垪瀛樺湪鍞竴鏋侀檺銆傝鏈夋暟鍒梄n , 鑻ュ瓨鍦∕>0,浣垮緱涓鍒囪嚜鐒舵暟n,鎭掓湁|Xn|<...
绛旓細锛屽嵆濡傛灉鏁板垪鏀舵暃锛鏋侀檺瀛樺湪锛屼笖鏋侀檺鍞竴銆
绛旓細1銆佸敮涓鎬э細鑻ユ暟鍒楃殑鏋侀檺瀛樺湪锛屽垯鏋侀檺鍊兼槸鍞竴鐨勶紝涓斿畠鐨勪换浣曞瓙鍒楃殑鏋侀檺涓庡師鏁板垪鐨勭浉绛锛2銆佹湁鐣屾э細濡傛灉涓涓暟鍒楁敹鏁涳紙鏈夋瀬闄愶級锛岄偅涔堣繖涓暟鍒椾竴瀹氭湁鐣屻備絾鏄紝濡傛灉涓涓暟鍒楁湁鐣岋紝杩欎釜鏁板垪鏈繀鏀舵暃銆備緥濡傛暟鍒1锛-1锛1锛-1锛屸︹︼紝(-1)n+1 锛屸︹3銆佷繚鍙锋э細鑻 锛堟垨<0锛夛紝鍒欏浠讳綍 m鈭堬紙...
绛旓細鏄厖鍒嗕笉蹇呰鏉′欢锛岃鎯呭鍥炬墍绀
绛旓細鏁板垪鏀舵暃鏄鏁板垪鐨勪竴涓ц川锛屾暟鍒楁瀬闄愬敮涓鏄竴涓懡棰樸備簩鑰呭叧绯绘槸杩欐牱鐨勶細濡傛灉鏁板垪鏀舵暃锛鍒欏繀鏈夋瀬闄愶紝杩欎釜鏋侀檺鏄敮涓鐨勶紱鍙嶈繃鏉锛屽鏋滄暟鍒鏈夋瀬闄愶紝鍒欐暟鍒楁敹鏁涖
绛旓細鎬诲瓨鍦ㄦ鏁存暟N锛屼娇寰楀綋m,n>N鏃讹紝瀵逛竴鍒噚鈭圖锛屾湁 璁炬暟鍒梴Xn}锛屽鏋瀛樺湪甯告暟a锛屽浜庝换鎰忕粰瀹氱殑姝ｆ暟q锛堟棤璁哄灏忥級锛屾诲瓨鍦ㄦ鏁存暟N锛屼娇寰梟>N鏃讹紝鎭掓湁|Xn-a|<q鎴愮珛锛屽氨绉版暟鍒梴Xn}鏀舵暃浜巃锛堟瀬闄愪负a锛夛紝鍗虫暟鍒梴Xn}涓鏀舵暃鏁板垪锛圕onvergent Sequences锛夈鏁板垪鏀舵暃<=>鏁板垪瀛樺湪鍞竴鏋侀檺銆
绛旓細1. 鏈夌晫鎬э細鏀舵暃鏁板垪蹇呭畾鏄湁鐣岀殑锛屽嵆瀛樺湪涓涓父鏁癕锛屼娇寰楄鏁板垪鐨勬墍鏈夐」閮藉皬浜庣瓑浜嶮銆2. 鍗曡皟鎬э細鏀舵暃鏁板垪鍙兘鏄崟璋冮掑鎴栧崟璋冮掑噺鐨勶紝涔熷彲鑳芥槸鏃笉鍗曡皟閫掑涔熶笉鍗曡皟閫掑噺鐨勩3. 鏋侀檺鍞竴鎬э細鏀舵暃鏁板垪鐨勬瀬闄愭槸鍞竴鐨勶紝鍗冲鏋滀竴涓鏁板垪鏀舵暃锛鍒欏叾鏋侀檺鏄敮涓鐨勩4. 淇濆彿鎬э細鑻ユ暟鍒鐨勯」閮藉ぇ浜庯紙鎴...
绛旓細鏈夋瀬闄愬張绉颁负鏀舵暃,鎵浠ユ湁鏋侀檺鐨勬暟鍒楀氨鏄鏀舵暃鏁板垪 鏈夌晫涓嶄竴瀹氭湁鏋侀檺,浣嗘湁鏋侀檺涓瀹氭湁鐣.
绛旓細鐢ㄦ瘮杈冪畝鍗曠殑绛変环鏃犵┓灏忔潵浠ｆ浛鍘熸潵澶嶆潅鐨勬棤绌峰皬鏉ュ 1/n * sin(1/n) 鐢1/n^2 鏉ヤ唬鏇 4銆鏀舵暃鏁板垪鐨鏋侀檺鏄鍞竴鐨勶紝涓旇鏁板垪涓瀹氭湁鐣岋紝杩樻湁淇濆彿鎬э紝涓庡瓙鏁板垪鐨勫叧绯讳竴鑷淬備笉绗﹀悎浠ヤ笂浠讳綍涓涓潯浠剁殑鏁板垪鏄彂鏁ｆ暟鍒椼傚彟澶栬繕鏈夎揪鏈楄礉灏旀敹鏁涘噯鍒,鏌タ鏀舵暃鍑嗗垯,鏍瑰紡鍒ゆ暃娉曠瓑鍒ゆ柇鏀舵暃鎬с
