如何利用幂级数展开求x的值?
x/sqrt(1+x^2)=x(1+x^2)^(-1/2),
利用(1+x)^a的幂级数展开式(1+x)^a=1+ax+[a(a-1)/2]x^2+[a(a-1)(a-2)/3!]x^3+...+[a(a-1)(a-2)...(a-n+1)/n!]x^3+...这里x换成x^2，a换成-1/2带入求出(1+x^2)^(-1/2)的幂级数
用2的答案乘以x得出最后答案
或者
x/sqrt(1+x^2)=x(1+x^2)^(-1/2)
(1+x^2)^(-1/2)用二项式定理展开
用2的答案乘以x得出最后答案
如果你熟悉二项式定理用第二个方法算更加方便
绛旓細ln3=1+ln锛3/e锛=1+ln(1+(3-e)/e),鐒跺悗鍒╃敤ln(1+x)灞曞紑寮忚绠,x=(3-e)/e
绛旓細1/(1-x)娉板嫆灞曞紑寮忚璇︾粏杩囩▼绛旀鏄1+x+x2+x3鈥︹1/(1-x)娉板嫆灞曞紑寮忚璇︾粏杩囩▼绛旀鏄1+x+x2+x3鈥︹︽嘲鍕掑睍寮寮忓張鍙骞傜骇鏁板睍寮娉 f(x)=f(a)+f'(a)/1!*(x-a)+f''(a)/2!*(x-a)^2+...+f(n)(a)/n!*(x-a)^n 鐜板湪f(x)=1/(1-x)閭ｄ箞姹傚寰楀埌f'(x)=-1/(1-x...
绛旓細浣嗘槸锛屾湁鍘熷嚱鏁板苟涓嶄唬琛ㄥ畠鑳藉鐢ㄥ熀鏈垵绛夊嚱鏁板舰寮忔潵琛ㄨ揪銆傛晠鎴戜滑鍙互鑰冭檻锛屼娇鐢ㄦ嘲鍕掑叕寮忓皢f(x)杩涜灞曞紑涓哄箓绾ф暟锛岃绠楀叾鏀舵暃鍩熷悗鍐嶈绠楀畠鐨勪笉瀹氱Н鍒嗐傗憼浣跨敤楹﹀厠鍔虫灄鍏紡瀵筬(x)=e^(x^2)杩涜閮ㄥ垎灞曞紑锛屽彲浠ユ敼鍐欎负涓涓箓绾ф暟銆傗憽鏍规嵁骞傜骇鏁扮殑鏀舵暃鍩熸眰娉曪細姹傗憼涓墍寰楀箓绾ф暟鐨勬敹鏁涘崐寰凴锛氬垯鈶犱腑骞傜骇鏁...
绛旓細璇︽儏濡傚浘鎵绀 鏈変换浣曠枒鎯戯紝娆㈣繋杩介棶
绛旓細濡傚浘鎵绀猴細鐣欐剰褰搉=0鏃剁涓椤规槸0锛屾墍浠ョ敱n=1寮濮嬩篃鍙互
绛旓細鍏堟壘鍑轰竴涓娇鏂圭▼涓よ竟姣旇緝鎺ヨ繎鐨x鐨勫t,鐒跺悗鑰冭檻鍑芥暟 f(x)=(c-algx)/b.閭ｄ箞鍘熸柟绋嬪氨鏄眰f(x)鐨勪笉鍔ㄧ偣鐨勮繎浼煎.鍙互鍒╃敤涓嶆柇鐨勮凯浠ｆ眰鍑哄厖鍒嗚繎浼肩殑x锛歺绾︾瓑浜巉(f(f(鈥︹(t))),杩唬鐨勬鏁拌秺澶,寰楀埌鐨剎鐨勫间篃灏辫秺绮剧‘.鏂规硶2锛氬皢lnx鐢═aylor鍏紡鍋骞傜骇鏁板睍寮,寰楀埌涓涓叧浜巟鐨勫椤瑰紡鏂圭▼...
绛旓細+ (kx)^5/5! - (kx)^7/7! + (kx)^9/9! - ...杩欐槸鍩轰簬姝ｅ鸡鍑芥暟鐨骞傜骇鏁板睍寮寮忥紝鍏朵腑 k 鏄父鏁般3. 骞傜骇鏁板睍寮寮忥細1/(1-kx)1/(1-kx) 鍙互灞曞紑涓哄箓绾ф暟锛屽叿浣撳睍寮寮忎负锛1/(1-kx) = 1 + kx + (kx)^2 + (kx)^3 + (kx)^4 + ...杩欐槸鍩轰簬鍑芥暟 1/(1-x) 鐨勫箓...
绛旓細楂樻暟璇炬湰涓婂鍑芥暟骞傜骇鏁扮殑灞曞紑寮忓敮涓鎬х殑浠嬬粛濡備笅鍥炬墍绀猴紝鏁欐潗涓婁篃鏈夎瘉鏄庤繃绋嬶紝璇佹槑鏂规硶鏄亣璁句笉鍞竴锛岀浉鍑忓緱闆跺彲瀵煎嚭鐭涚浘锛屾晠鍞竴銆備緥瀛愭暀鏉愪笂涔熸湁锛岃瘉鏄庤繃绋嬪拰渚嬪瓙澶繃澶嶆潅锛屼笉鑳芥墦鍑烘潵锛屾湁闇瑕佺殑璇疯嚜琛屾煡鐪嬫暀鏉愩
绛旓細骞傜骇鏁板睍寮寮忎腑锛屽悗闈㈢殑x鐨鑼冨洿鏄敹鏁涘煙銆傚箓绾ф暟鐨勬敹鏁涘煙鏄互鍘熺偣涓轰腑蹇冪殑鍖洪棿銆 璁綬涓烘敹鏁涘崐寰勶紝鍒 锛-R,R锛夊唴骞傜骇鏁版敹鏁涳紝鈭鈭>R鐨剎 骞傜骇鏁版槸鍙戞暎鐨勶紝 x=卤R 鍙兘鏀舵暃涔熷彲鑳藉彂鏁ｇ殑銆
绛旓細瑙ｏ細f(x)=(3x-5)/[(x-3)(x-1)]=A/(x-1)+B/(x-3)A(x-3)+B(x-1)=(A+B)x-3A-B=3x-5; 瀵规瘮绯绘暟:A+B=3,-3A-B=-5; 寰楋細2A=2锛孉=1锛汢=2锛沠(x)=1/(x-1)+2/(x-3)=-1/(1-x)-(2/3)*1/(1-x/3)=-[1+(-x)]^(-1)-(2/3)[1+(-x/3)]^...
