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1Taylor21Taylor2**3.f(x)x0O(x0,r)f(x)x0Taylor*).,(,)(!)()(0000)(rxOxxxnxfxfnnn∈−=∑∞=(1)f(x)=ex=∑∞=0!nnnx!!3!2132nxxxxn++++=+,x(,+)(2)f(x)=sinx=∑∞=++−012!)12()1(nnnxn)!12()1(!5!31253+−+−+−=+nxxxxnn+,x(,+)1(3)f(x)=cosx=∑∞=−02!)2()1(nnnxn)!2()1(!4!21242nxxxnn−+−+−=+,x(,+)(4)f(x)=arctanx=∑∞=−−−−112112)1(nnnxn12)1(531253+−+−+−=+nxxxxnn+,x[1,1](5)f(x)=ln(1+x)=∑∞=+−11)1(nnnxnnxxxxxnn1432)1(432−−++−+−=+,x(1,1](6)0fxx()()=+1ααmf(x)=(1+x)m=1+mx+22)1(xmm−+++x1−mmxmx(,+)α0∑∞=α⎟⎟⎠⎞⎜⎜⎝⎛α=+0)1(nnxnx,⎪⎩⎪⎨⎧−−≤−∈−∈−∈.0,01,1],1,1[],1,1(),1,1(αααxxx=⎟⎟⎠⎞⎜⎜⎝⎛nα!)1()1(nn+−α−αα,(n=1,2,)10=⎟⎟⎠⎞⎜⎜⎝⎛αf(x)x0O(x0,r)O(x0,r)*1122531)(xxxf−+=0=x⎟⎠⎞⎜⎝⎛+⋅+⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅=xxxf21172311211)(61−=α()nnnnxxf∑∞=++⎥⎦⎤⎢⎣⎡−−=01123171)().21,21(−∈x2xxf3sin)(=6π=x)6(3cos41)6(6sin433sin41sin43sin)(3πππ−−⎟⎠⎞⎜⎝⎛−+=−==xxxxxxf2)6(3cos41)6cos(83)6sin(833πππ−−−+−=xxx23).,(,)6)(132()!2()1(83)6()!12()1(833)(2120012+∞−∞∈−−⋅−−−+−=−∞=∞=+∑∑xxnxnxfnnnnnnnππ3)0(,ln)(=xxxf11+−xx,11+−=xxt)10(,11−+=tttx5)1ln()1ln(11lnlnttttx−−+=−+=nnnnntntn∑∑∞=∞=++−=1111)1(.0,)11(12121212112121+−⋅+=⋅+=∑∑∞=++∞=xxxntnnnnn2421)(xxf=1=x∑∞=−=−+==0)1()1(111)(nnxxxxg).2,0(,)1)(1()1()(')(101∈−+=−=−=∑∑∞=∞=−xxnxnxgxfnnnn5f(x)=arcsinx0=x(6))21(−=αx∈(1,1)211x−=212)1(−−x=∑∞=−⎟⎟⎠⎞⎜⎜⎝⎛−0221)(nnxn=1+221x+483x++nxnn2!)!2(!)!12(−+0x∫−xtt021d=arcsinxarcsinx=x+∑∞=++−11212!)!2(!)!12(nnnxnnx[1,1]x=1Raabex=12π=1+∑∞=+⋅−0121!)!2(!)!12(nnnn3)()(xgxf)()(xgxfCauchyf(x)R∑∞=0nnnxa1g(x)∑,∞=0nnnxb3R2f(x)g(x)Cauchyf(x)g(x)=()()=,∑∞=0nnnxa∑∞=0nnnxb∑∞=0nnnxccn=,∑=−nkknkba0∑∞=0nnnxc≥RminR1R2b00)()(xgxf)()(xgxf=,∑∞=0nnnxc(∑)()=,∞=0nnnxb∑∞=0nnnxc∑∞=0nnnxaxb0c0=a0⇒c0=00ba,b0c1+b1c0=a1⇒c1=0011bcba−b0c2+b1c1+b2c0=a2⇒c2=002112bcbcba−−cnexsinx(x5)exsinx=(!4!3!21432xxxx+++++)(−+−!5!353xxx)=x+53230131xxx−++xexsin∞=Rx(,+)tanx(x5)tanxtanx=xxcossin=c1x+c3x3+c5x5+(c1x+c3x3+c5x5+)(−+−!4!2142xx)=−+−!5!353xxxx,x3x5c1=1,c3=31,c5=152,tanx=x+31x3+152x5+447u−11==1+u+u∑∞=0nnu2+u=+−!4!242xxxcos1=1+(+−!4!242xx)+(+−!4!242xx)2+=1+x2+245x4+sinxxcos1Cauchytanxx=tanx(0x2π,2π)ef(x)ln(1+f(x))8(xxexfsin)(=0=x4)+−=+−==+∞=∑6)!12()1(sin3120xxxnxunnn+++++===∑∞=xxxxnxexfnnx4320sinsin241sin61sin21sin1!sin)(),(,81211)(42sin+∞−∞∈+−++==xxxxexfxxexfcos)(=0=x−+−==42241211cosxxxu∑∞==0!cos)(nnnxxflnxxsin(x4)xxsinf(x)=⎪⎩⎪⎨⎧=≠.01,0,sinxxxxsinxxxsin=−+−!5!3142xxu=−+−!5!342xxln(1+u)=u−+3232uu5lnxxsin=(−+−!5!342xx)21(−+−!5!342xx)2+=−−−180642xx9xxsin=∏∞=π−1222)1(nnx,ln)1(222π−nxlnxxsin=∑∞=π−1222)1ln(nnx=∑∞=+π+π1444222)21(nnxnxx2x4∑∞=121nn=62π,∑∞=141nn=904πlnxxsinx6x8∑∞=161nn∑∞=181nn1x)(xf0x0Taylor*0xx=Taylor)(xf0x)(xf)(xf0xx=Taylor0x0x2**396