若一个无穷等比数列的公比q满足|q|<1,且每一项都等于它以后各项和的k倍,则k的取值范围是
\u82e5\u4e00\u4e2a\u65e0\u7a77\u7b49\u6bd4\u6570\u5217\u7684\u516c\u6bd4q\u6ee1\u8db3|q|<1,\u4e14\u6bcf\u4e00\u9879\u90fd\u7b49\u4e8e\u5b83\u4ee5\u540e\u5404\u9879\u548c\u7684k\u500d,\u5219k\u7684\u53d6\u503c\u8303\u56f4\u662f\u7b2cn\u9879\u662faq^(n-1)
\u4ed6\u7684\u540e\u9762\u662f\u7b49\u6bd4\u6570\u5217\uff0c\u9996\u9879\u662faq^n,\u516c\u6bd4\u662fq
\u6240\u4ee5\u548c=aq^n/(1-q)
\u6240\u4ee5aq^(n-1)/[aq^n/(1-q)]=k
(1-q)q=k
k=q-q^2=-(q-1/2)^2+1/4
-1<q<1, \u4e14 q\u4e0d\u7b49\u4e8e0
\u6240\u4ee5q=1/2,k\u6700\u5927=1/4
q=-1\uff0ck\u6700\u5c0f=-2
q=0\uff0ck=0,k=-q^2+q=0,q\u8fd8\u53ef\u4ee5\u7b49\u4e8e1\uff0c\u4f46\u4e0d\u7b26\u5408|q|<1
\u7efc\u4e0a
-2<k<0,0<k<1/4
a1=k*a2/(1-q),-1o\u6216k<\uff0d2
am=k[a(m+1)+a(m+2)+.......]am=ka(m+1)/(1-q)
am=k*am*q/(1-q)
k=(1-q)/q
=1/q -1
0<|q|<1
所以 1/q<-1或1/q>1
所以 k<-2或k>0
绛旓細鎵浠 1/q<-1鎴1/q>1 鎵浠 k<-2鎴杒>0
绛旓細a1/(1锛q)=6锛屼笖a1²/(1锛峲²)=18锛屾眰鍑簈=1/3銆
绛旓細瑙ｇ瓟锛鏃犵┓绛夋瘮鏁板垪鏈夊悇椤瑰拰锛屽垯0<|q|<1 鍒╃敤鍏紡S=a1/(1-q)鍒 am=k*am*q/(1-q)鈭 k=(1-q)/q=1/q-1 鈭-1<q<0鎴0<q<1 鈭 1/q<-1鎴 1/q>1 鈭 1/q-1< - 2鎴 1/q-1>0 鍗砶<-2鎴杒>0 鈭 瀹炴暟k鐨勫彇鍊艰寖鍥存槸k<-2鎴杒>0 ...
绛旓細浠栫殑鍚庨潰鏄绛夋瘮鏁板垪锛岄椤规槸aq^n,鍏瘮鏄q 鎵浠ュ拰=aq^n/(1-q)鎵浠q^(n-1)/[aq^n/(1-q)]=k (1-q)q=k k=q-q^2=-(q-1/2)^2+1/4 -1<q<1, 涓 q涓嶇瓑浜0 鎵浠=1/2,k鏈澶=1/4 q=-1锛宬鏈灏=-2 q=0锛宬=0,k=-q^2+q=0,q杩樺彲浠ョ瓑浜1锛屼絾涓嶇鍚坾q|<1 ...
绛旓細鏃犵┓绛夋瘮鏁板垪鐨勫拰S=a1*锛1-q^n)\(1-q)鍥犱负q鏄釜缁濆鍊煎皬浜庨浂鐨勬暟,鎵浠ュ綋n瓒嬩簬鏃犵┓鐨勬椂鍊,q^n瓒嬩簬闆,鎵浠涓烘棤绌锋椂鍜孲=a1\(1-q),a1\(1-q)-Sn=1\k*a1*q^(n-1),鏈鍚庡寲绠鎴 q=1\(k+1),鍐嶅埄鐢▅q|
绛旓細{k|k<-2鎴杒>0}銆傚洜涓烘槸鏃犵┓绛夋瘮鏁板垪锛屼笖0<|q|<1锛屾墍浠n+1+An+2+An+3+鈥︹=An+1/(1-q)锛岃孉n+1=Anq锛屼粠鑰屽緱1=kq/(1-q)锛屾眰寰梜=(1/q)-1銆 鍙1/q<-1鎴1/q>1锛屽洜姝わ紝k<-2鎴杒>0銆
绛旓細鏃犵┓绛夋瘮鏁板垪鍙湁褰撳叕姣旓副q锔憋紲1鎵嶈兘鐢ㄥ叕寮忔眰.渚嬪涓鑸殑鏃犵┓绛夋瘮鏁板垪锛歛1锛宎1q锛宎1q2锛屸︼紝a1qn-1锛屸.褰擄副q锔憋紲1鏃讹紝n鈫掆垶鏃跺彲寰條imSn=lim a1锛1-qn锛/1-q =锛坅1/1-q锛壜穕im锛1-qn锛=锛坅1/1-q锛夛紙lim 1-lim qn锛=a1/1-q 鎶妉im Sn锛坣鈫掆垶锛夊彨杩欎釜鏃犵┓绛夋瘮鏁板垪锛鍏瘮q婊¤冻锔...
绛旓細褰q=1鏃讹紝鏃犵┓绛夋瘮鏁板垪{an}涓哄父鏁板垪锛岃sn涓哄叾鎵鏈夐」涔嬪拰锛屽垯sn=na1=23锛屽嵆a1=23n锛屽綋n鈫+鈭炴椂锛宎1鈫0锛屽嵆a1锛0锛涘綋-1鈮锛1鏃讹紝sn=a1(1-qn)1-q锛屽綋n鈫+鈭炴椂锛宷n鈫0锛屼簬鏄湁sn=a11-q=23锛屽嵆a1=23锛1-q锛夛紝褰搎=-1鏃讹紝a1鏈澶э紝鎵浠ュ緱鍒0锛渁1鈮34锛屽張q鈮0锛屽緱鍒癮1鈮23...
绛旓細璁鏃犵┓绛夋瘮鏁板垪 鐨勫叕姣涓q锛屼笖 锛 琛ㄧず涓嶈秴杩囧疄鏁 鐨勬渶澶ф暣鏁帮紙濡 锛,璁 锛屾暟鍒 鐨勫墠 椤瑰拰涓 锛屾暟鍒 鐨勫墠 椤瑰拰涓 .锛堚厾锛夎嫢 锛屾眰 锛涳紙鈪★級鑻ュ浜庝换鎰忎笉瓒呰繃 鐨勬鏁存暟n锛岄兘鏈 锛岃瘉鏄庯細 .锛堚參锛夎瘉鏄庯細 锛 锛夌殑鍏呭垎蹇呰鏉′欢涓 . 锛堚厾锛 锛涳紙鈪★級绛...
绛旓細Sn=a1(1-q^n)(1-q),褰搉瓒嬩簬鏃犵┓鏃讹紝浠呭綋|q|<1鏃舵敹鏁涳紝鏀舵暃浜巃1/(1-q)=q 寰:a1=q(1-q)<=(q+1-q)²/4=1/4, 褰搎=1-q, 鍗硄=1/2鏃跺彇鏈澶у 鍥犳a1鏈澶у间负1/4
