如何证明数列无穷小？
\u8bc1\u660e\u6570\u5217\u4e3a\u65e0\u7a77\u5c0f\u5217\u7531\u4e8e lim |a(n+1|/|a(n)| = c\uff0c
\u6839\u636e\u6781\u9650\u7684\u5b9a\u4e49\uff0c\u53d6 \u03b5\uff1d(1\uff0dc)/2 ( \u03b5 >0 )
\u5fc5\u7136\u5b58\u5728 N\uff0c \u5f53 n>N\u65f6\u6709\uff1a
| |a(n+1)|/|a(n)| \uff0d c | \uff1c \u03b5 (n>N)
==> | |a(n+1)|/|a(n)| \uff0d c | \uff1c (1\uff0dc)/2 (n>N)
==> |a(n+1)|/|a(n)| \uff1c (1\uff0dc)/2 + c (n>N)
==> |a(n+1)|/|a(n)| \uff1c (1 + c)/2 \uff1c 1 (n>N)
==> |a(n+1)| N)
\u4e5f\u5c31\u662f\u8bf4n>N\u65f6\uff0c|a(n)|\u662f\u4e2a\u5355\u8c03\u9012\u51cf\u6570\u5217\uff0c\u4e14\u6709\u4e0b\u754c0\uff0c\u56e0\u6b64 |a(n)|\u5fc5\u6709\u6781\u9650.
\u4e8e\u662f\u6839\u636e\uff1a lim |a(n+1|/|a(n)| = c
==> lim |a(n)| \uff1d0 (\u5982\u679c\u4e0d\u4e3a0\uff0c\u5219\u4f1a\u5f97\u51fac=1\uff0c\u4e0e\u9898\u8bbec<1\u77db\u76fe)
\u7531 lim |a(n)| \uff1d0
==> lim a(n) \uff1d 0
\u5373a(n)\u4e3a\u65e0\u7a77\u5c0f\u5217\u3002
\u8981\u70b9: 1. \u6536\u655b\u7684\u5e8f\u5217\u5fc5\u5b9a\u6709\u754c
2. \u6536\u655b\u7684\u5e8f\u5217"\u6700\u591a\u53ea\u6709\u6709\u9650\u9879\u79bb\u6781\u9650\u6bd4\u8f83\u8fdc"
\u4efb\u53d6e>0, \u5b58\u5728N1>0\u4f7f\u5f97\u5f53n>N0\u65f6|an|<e/2
(\u5f53n\u975e\u5e38\u5927\u65f6\u540e\u9762\u7684\u9879\u90fd\u5f88\u5c0f, \u5e73\u5747\u503c\u4e5f\u5e94\u8be5\u5f88\u5c0f)
\u518d\u53d6M=max{|a1|,|a2|,...,|aN0|}, \u4ee5\u53caN=[2MN0/e]+1
\u90a3\u4e48n>N\u65f6|a1+a2+...+aN0|/n < e/2
(\u524d\u9762\u6709\u9650\u9879\u6bd4\u8f83\u5927\u7684\u88ab\u63a7\u5236\u4f4f\u4e86)
\u800c|a(N0+1)+ ...+an|/n < e/2
\u6240\u4ee5|(a1+...+an)/n| < e
\u5373\u4e3a\u7ed3\u8bba
数学归纳法
绛旓細(n>N)涔熷氨鏄n>N鏃讹紝|a(n)|鏄釜鍗曡皟閫掑噺鏁板垪锛屼笖鏈変笅鐣0锛屽洜姝 |a(n)|蹇呮湁鏋侀檺.浜庢槸鏍规嵁锛 lim |a(n+1|/|a(n)| = c ==> lim |a(n)| 锛0 (濡傛灉涓嶄负0锛屽垯浼氬緱鍑篶=1锛屼笌棰樿c<1鐭涚浘)鐢 lim |a(n)| 锛0 ==> lim a(n) 锛 0 鍗砤(n)涓鏃犵┓灏鍒椼
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绛旓細濡傚浘
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绛旓細x=0.99999999鈥︹10x=9.9999999鈥︹10x-x=9 x=1 鎵浠(n-->oo) lim xn =1 (n-->oo) lim xn-1=0 {xn}鏄鏃犵┓灏
绛旓細璇佹槑鏁板垪鏄鏃犵┓灏忛噺 鎴戞潵绛 1涓洖绛 #鐑# 宸插濂虫у氨搴旇鎵挎媴瀹堕噷澶ч儴鍒嗗鍔″悧?dduwayne 2015-08-06 路 TA鑾峰緱瓒呰繃830涓禐 鐭ラ亾灏忔湁寤烘爲绛斾富 鍥炵瓟閲:593 閲囩撼鐜:0% 甯姪鐨勪汉:597涓 鎴戜篃鍘荤瓟棰樿闂釜浜洪〉 鍏虫敞 灞曞紑鍏ㄩ儴 鏇村杩介棶杩界瓟 杩介棶 璋㈣阿,澶у笀 鏄庣櫧浜 鏈洖绛旂敱鎻愰棶鑰...
绛旓細n!/n^n =1/n*2/n*...*n/n <1/n 鑰1/n鏋侀檺鏄0,鎵浠!/n^n鏋侀檺鏄0,鍗冲畠鏄鏃犵┓灏
绛旓細3.鏈鍚庯紝鎴戜滑闇瑕璇佹槑鍘熸暟鍒楃殑鏋侀檺鏄敮涓鐨勩傝繖閫氬父鍙互閫氳繃璇佹槑濡傛灉鏈変袱涓笉鍚岀殑鏋侀檺锛岄偅涔堣繖涓や釜鏋侀檺瀵瑰簲鐨勯珮闃鏃犵┓灏忔暟鍒鏄笉鍚岀殑鏉ュ疄鐜般傚叿浣撴潵璇达紝濡傛灉鎴戜滑鏈変竴涓暟鍒梴an}锛屾垜浠兂瑕佽瘉鏄巐im(n->鈭)an=L銆傛垜浠彲浠ユ壘鍒颁竴涓暟鍒梴bn}锛屼娇寰楀浜庢墍鏈夌殑n锛宐n鏄痑n鐨勪竴涓猭闃鏃犵┓灏忛噺锛屽嵆|bn|鈭)...
绛旓細鏃犵┓灏忔暟鍒灏辨槸褰撻」鏁皀鏃犵┓澶ф椂锛宎n瓒嬭繎浜0鐨勬暟鍒楋紝瀹冩湁濂藉绉嶅舰寮忥紝渚嬪锛1 銆1/2銆1/3 銆1/4 銆佲︹1/n 1/3 銆1/5 銆1/7 銆1/9 銆佲︹1/锛2n+1锛夎嫢{an}鏄棤绌峰皬鏁板垪锛屾敼鍙榹an}涓殑鏌愭湁闄愰」涔嬪悗锛屽畠浠嶆槸鏃犵┓灏忔暟鍒椼
