1. Ряды (853737), страница 4
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; + . . "" x0 ", : ; " f (x) x0, , f (x)? - , 0 .. 7.1. &. . ; "8< ;1=x x 6= 0,f (x) = : e(7:7)0 x = 0:< F " "" x,f (0) = f 0 (0) = : : : = f (n) (0) = : : : = 0255. . 0"" ; " x0 = 0 . 8., ; 0 " x0 = 0 x , " . >; , , " f (x), " (7.7), ; x0 = 0. @ , " f (x) ;. 7.2. f (x) **+ x0 *+. , *+ f (x) & !, , * !00 (x0)f0f (x) = f (x0 ) + f (x0)(x ; x0) + 2! (x ; x0)2 + : : : +(n)f+ : : : + n(!x0 ) (x ; x0)n + Rn(x)(7.8) , n ! 1:nlim!1 Rn (x) = 0:(7:9) .
@.. . f (x) ; x0, . . + (7.6). % , . + f (x) n- (7.6), , (7.8), Rn(x) " ;, , n ! 1 x0.F.. +, (7.9):nlim!1 Rn(x) =230 (x0 )(n)ff(x)0n 5 =4f (x0 ) += nlimf(x);(x;x)+:::+(x;x)00!1 1!n!= nlim!1 jf (x) ; Sn(x)j = 0 Sn(x) { n- (7.6). /, ; (7.6) f (x), . f (x) ;. ut568 + " ; .# +.
7.3. x0 *+ f (x) " ". " M > 0, jf (n)(x)j M(7:10)" n = 0 1 2 : : : x x0 , *+ f (x) ! & . . 8 7.2, ., Rn (x) " ; (7.8) , n ! 1 x0. , , " N+ . Rn(x), (7.10) : (n+1)n+1 f(x+(x;x))Mjx;x000jn+1 jRn(x)j = (x ; x0) (n + 1)! (7:11)(n + 1)! 0 < < 1.8 . F ., 1 M jx ; x0jn+1X, 0, n=0 (n + 1)!, 0 , n ! 1:M jx ; x0jn+1 ! 0:(n + 1)!/, e (7.11),nlim!1 Rn (x) = 0:7 .
.. ut7 . , x0 = 0 " f (x) x:(n)00 (0) 2ff0(7:12)f (x) = f (0) + f (0)x + 2! x + : : : + n!(0) xn + : : : :% 5 " f (x).57* . -( /; 7.3 + .I. % " ex, sin x,cos x (;R R). F.,j(ex)(n) j = jexj eRCj(sin x) j(n)j(cos x)(n) j ! = sin x + n 2 1C ! = cos x + n 2 1:&, " ex, sin x cos x ; (n)f x. , 0"" ; n(0)0 ", ! +2nex = 1 + x + x2! + : : : + xn! + : : : C(7:13)52n+13xxxnsin x = x ; 3! + 5! ; : : : + (;1) (2n + 1)! + : : : C(7:14)242ncos x = 1 ; x2! + x4! ; : : : + (;1)n (2xn)! + : : : :(7:15)II. - + 6.1 " f (x) = ln(1+ x) f (x) = arctg x $ :32xx(7:16)ln(1 + x) = x ; 2 + 3 ; : : : x 2 (;1 1)C35xxarctg x = x ; 3 + 5 ; : : :(7:17) x 2 (;1 1).
7.1. + (7.16) x = 1.8.,ln 2 = 1 ; 12 + 31 ; 14 + : : : :(7:18)58 7.2. + (7.17) + x = 1. 0 = 1; 1 + 1 ; 1 + ::: :(7:19)43 5 7+ (7.18) (7.19) + .. ln 2 . . , + 0 + . (7.18) (7.19). 0, N, #. # .*#+, "!,1. , %"< ( ) 0 , - F %"< ( ) (. " 7.1).2. , %"< ( ) 0 , - (. " 7.1).3. $ F %"< ( ) %"< ( )?4.
$%"" " %"< F (. " 7.2).5. $%"" " %"< F (. " 7.3).6. 7 D %"< x, sin , cos , ln(1 + ) arctg .f xxf xf xxf xe3. B , (. 7.1).59xxf xxx 8 <8> - @/0>/2 202.@/2 * <8>, + , Lf + P # f . F, # f + . " '1 '2 ::: 'n ::: .;, L , + . ., " ". @, . + + Ak sin (!k t + k ), !k { .. . , 3..* + $ 8.1. 3b' (x) (x) ZG a b ], ' (x) (x) dx = 0:a 8.2. 8 G a b ] " '1 (x) '2 (x) : : : 'n (x) : : : G a b ], :) " 0 . G a b ],Zb..
'n (x) 'm (x) dx = 0 n 6= m n m = 1 2 : : : C aZb) 'n2 (x) dx 6= 0 n = 1 2 : : : :aZbE 0 'n2 (x) dx = 1 (n = 1 2 : : :), "a'1 (x) '2 (x) : : : 'n (x) : : : G a b ].60 8.3.. '1 (x) '2 (x) : : : 'n (x) : : : { . G a b ] " . f (x) { Zb G a b ] ", .. j f (x) j dx .a; Zbf (x) 'n (x) dxa n = 1 2 : : :(8:1)cn = Zb'n2 (x) dxa &**+ 6 " f (x) f'n (x)g. 1Xcn 'n(x)" n=1 6 " f (x) "f'n(x)g. .1Xf (x) cn 'n(x)(8:2) n=1(8:3) , (8.2) 3. " f (x) . f'n(x)g, .. 0"" cn " (8.1).
8.1 (!#"!#+ $ < +). " *+ f (x) G a b ] " f'n (x)g, ..f (x) =1Xn=1cn 'n(x) x 2 G a b ]:(8:4)! (8.4) 6 *+ f (x) " f'n(x)g, .. &**+ " * (8.1). . ; " 'm (x) m 2 NII, G a b ], ,# + 0 . E ". +. ", . (0 61 ). 0, + (8.4) " 'm (x),1Xf (x) 'm (x) = cn 'n(x) 'm (x) x 2 G a b ] m 2 NII (8:5)n=1 (8.5) G a b ].; ". + .
( ". ), (8.5) , Zb1 ZbXf (x) 'm (x) dx = cn 'n(x) 'm (x) dx m 2 NII:an=1a&. .. f'n(x)g, RbZbZb 2f (x) 'm (x) dx m 2 NIIf (x) 'm (x) dx = cm 'm (x) dx () cm = a Rb2aa' (x) dxa m.. " (8.1) . ut + " )(1C cos l xC sin l xC : : : C cos nl xC sin nl xC : : : (8:6) .# 2 l (. l { ." +. ). 8.2.
! (8.6) G ;l l ]. . 8 8.2 . . :Zl1 cos nl x dx = 0 n = 1 2 : : : C(8:7);l62ZlZl;l;lcos nl x cos ml x dx = 0 n 6= m n m = 1 2 : : : CZl;lZl;l1 sin nl x dx = 0 n = 1 2 : : : Ccos nl x sin ml x dx = 0 n m = 1 2 : : : Csin nl x sin ml x dx = 0 n 6= m n m = 1 2 : : : C(8:8)(8:9)(8:10)(8:11)Zln1 dx = 2 lC cos l x dx = sin2 nl x dx = l n = 1 2 : : : :;l;l;l(8:12)-+ # (8.7) { (8.12) .F+, , # (8.9) (8.12).&. " , n 6= m!ZlZl n + mnm1n;mcos l x cos l x dx = 2 cos l x + cos l x dx =;l;lZl2Zl2 l 1 lln+mln;m1= 2 (n + m) sin l x ;l + 2 (n ; m) sin l x ;l = 0.
sin k = 0 k. 8., # (8.9) . &. " , !Zl Zl 2 n 12nsin l x dx = 2 1 ; cos l x dx =;l;l!l1l2n= 2 x ; 2 n sin l x ;l = l:631 , Zlcos2 nl x dx = l;l.. # (8.12) . ; . ut 8.3 , 3.. 8.4. .l f (x) { Z G ;l l ] ", .. j f (x) j dx . ; ;llZZl11a0 = l f (x) dxCan = l f (x) cos nl x dxC;l;l(8.13)Zl1nbn = l f (x) sin l x dx n = 1 2 : : :;l &**+ 6 " f (x) " (8.6) " 2 l, !1nna0 + X(8:14)2 n=1 an cos l x + bn sin l x 6 " f (x) (8.6). .!1Xnna0(8:15)an cos l x + bn sin l x f (x) 2 +n=1 , (8.14) 3. " f (x) (8.6), .. 0"" a0 an bn " (8.13).@ 8.1 8.3. 8.3 (!#"!#+ #6#A!46 $< +).
" *+ f (x) G ;l l ] " (8:6), ..!1Xnna0an cos l x + bn sin l x x 2 G ;l l ]:f (x) = 2 +n=164! & 6 *+ f (x) " (8:6), .. &**+ " * (8.13). 8.1. 3. !N Xnn0T N (x) = 2 + n cos l x + n sin l xn=1 2 l (8.6) 0 , ..0"" 3. " T N (x) (8.6) 8<an = : 0n 8<bn = : 0n n = 0 1 : : : Nn > NCn = 1 : : : Nn > N: 8.1. @ 0"" 3. . 3."8<; l x < 0f (x) = : ;xx+;ll 0 x l: (8.6).< &. " (8.13) , 0"" 3." f (x) (8.6):ZlZ0Zl111a0 = l f (x) dx = l (;x ; l) dx + l (x + l) dx =0;l;l0 1 0 ;l 2 + 4 l 2 ; l 212 2 = lC= ; 2 l (x + l) ;l + 2 l (x + l) ;l =2lZlZ0n11an = l f (x) cos l x dx = ; l (x + l) cos nl x dx+;l;l 0Zln1ln1+ l (x + l) cos l x dx = ; l n (x + l) sin l x ;l +0 l 1Z0 n 1l1ln+ l n sin l x dx + l n (x + l) sin l x 0 ; l ;l0lZl n l n sin l x dx = ; n2l2 cos nl x ;l + n2l2 cos nl x 0 =065n ; 1]l2lG(;1)= n2 2 (;1 + cos n + cos n ; 1) =Cn2 2ZlZ0n11bn = l f (x) sin l x dx = ; l (x + l) sin nl x dx+;l;ll 0Z1n1ln+ l (x + l) sin l x dx = l n (x + l) cos l x ;l ;0 l 1Z0 n l1ln1; l n cos l x dx ; l n (x + l) cos l x 0 + l ;lnZll n cos nl x dx = nl ; n2 l cos n + nl = 2 l G 1 ;n (;1) ] :0&,8>0 n = 2 k<4la0 = lC an = > ;: (2 k ; 1)2 2 n = 2 k ; 1 k = 1 2 : : : (8.16)8>0n=2k<bn = > ; 4 l n = 2 k ; 1 k = 1 2 : : : :: (2 k ; 1) , 0"" 3.
(8.16) " f (x) (8.6), + . 3. 0 " (8.6):! l 4l1Xann0f (x) 2 +an cos l x + bn sin l x = 2 + n=1(8.17)101X @; (2 k ;1 1)2 cos (2 k ;l 1) x + 2 k1; 1 sin (2 k ;l 1) xA : >k=1, , + " (8.15), (8.17) "" . "=" . 8.2. @ 0"" 3. . 3."f (x) = sin x + sin2 x + 4 sin 3 x: (8.6) 2 .66< l = (8.6) f1C cos xC sin xC : : : C cos n xC sin n xC : : :g :,.#. " 2 sin2 x = 1 ; cos 2 x, f (x) = 12 + sin x ; 21 cos 2 x + 4 sin 3 x:; f (x) { 2 , , 8.1, 3. f (x) 2 0 . >0 +!, G ;l l ] "-, 3. . + G ;l l ] ,., , G ;l l ]. " " F J- 1876 ., 0 " " , , .
3. (8.6). 8.5. 3 f (x) -" G ;l l ], + , .@, x0 " f (x) " , 0 f (x). f (x0 ; 0) = x!limf (x) f (x0 + 0) = x!limx ;0x +000 8.4 ( !B#7 "7 ! !#). *+ f (x) " G ;l l ], & -" " f 0 (x) f (;l) = f (l). ! 6 *+ f (x) " (8:6) f (x) G ;l l ] , , " !1Xnna0an cos l x + bn sin l x x 2 G ;l l ]:f (x) = 2 +n=167 8.1. E " f (x) 8.4, S (x) 3., ", + f (x) ( 2 l) ., 8.4 . 3. S (x) .F ", . 8.6.
3 f (x) - G a b ], 0 + . , + f (x) ( ). 8.5 (34 ). *+ f (x) -" - G ;l l ], 6 " (8:6) G ;l l ], " !1a0 + Xnn2 n=1 an cos l x + bn sin l x =8>f (x) x ! f (x) " #>>>>>>< f (x ; 0) + f (x + 0) x ; # #$"# f (x)=>2>>>>f (;l + 0) + f (l ; 0) x = l:>>:(8:18)2 8.2. . " f (x) - G ;l l ] . x1 x2 : : : xs x = l :f (xk ) = f (xk ; 0) +2 f (xk + 0) k = 1 2 : : : nC(8.19)f ( l) = f (;l + 0)2+ f (l ; 0) :68; 3.
" f (x) (8:6) + G ;l l ]:!1Xnna0an cos l x + bn sin l x ; l x l (8:20)f (x) = 2 +n=1 a0 an bn { 0"" 3. " f (x) (. " (8.13)). - , 3. (8.19) + ", + f (x) ( 2 l) .. 8.3. '.8 , "<; l x < 0f (x) = : ;xx+;ll 0 x l 8.5. F. x = l , 3. (8:6) f (x) G ;l l ].. " S (x) 3. " (8.6) (;1 +1). ,. S (100 l)C S (151 l)C S (2245 l)C S (3475 l).< @, 3.
" (8.6) 8.1 (. " (8.17)). ; f (x) G ;l l ] (;l 0), (0 l), 8.5 3. (8.17) f (x) . , x = 0 3. (8.17) f (0 ; 0) + f (0 + 0) = ;l + l = 022 G ;l l ] { f (;l + 0) + f (l ; 0) = 0 + 2 l = l:22; , . " 8 :>;x ; l ; l x < 0>>< x + l 0 x lf (x) = > 0 x = 0>>:l x = l69( " + 8.1), 3. (8.17) 0 " . + G ;l l ]:10X4llf (x) = 2 + @; (2 k ;1 1)2 cos (2 k ;l 1) x +k=11(2k;1)1xA ; l x l:+ 2 k ; 1 sinl.
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