, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x

1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d n(x)) lim dx f n(x) 1

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x) x 1 n n x f n (x) = ε δ (a, b) {f n (x)} (a, b) f(x) x (a, b), ε >, n = n (x, ε), n n, f n (x) f(x) < ε 1.2 1 lim f n (x) = f(x) x n x 2

n n n f n (x) f(x) x (a, b), ε >, n = n (ε), n n, f n (x) f(x) < ε n = n (x, ε) n = n (ε) (a, b) x {f n (x)} f(x) lim f n (x) = f(x), x (a, b), f n (x) f(x), x (a, b) lim f n (x) = f(x), x (a, b), f n (x) f(x), x (a, b) ( 2 (i) (ii) b lim a f n (x) f(x), f n (x) f(x), x (a, b) f n (x)dx = b a ( lim f n (x))dx d dx f n(x) g(x), x (a, b), d f(x) = g(x) dx {f n (x)} f n (x) = x n, x 1 3

lim f n(x) = f(x) = {, x < 1 1, x = 1 1 f(x)dx =. 1 f n (x)dx = [ xn+1 n + 1 ]x=1 x= = 1 n + 1, n 1 lim f n (x)dx = 1 ( lim f n (x))dx lim f n (x) = f(x) lim x n = < ε x n ϵ x 1 x n ε n log x log ε n log ε log x n ε x 3 (i) b f n (x) < (, x (a, b), a ( f n (x))dx = b a f n (x)dx 4

(ii) d f n (x) <, dx f n(x) < (, x (a, b), d dx ( f n (x)) = d dx f n(x) 4 (i) (ii) f n (x) < f n (x) < f n (x) < (, x (a, b), f n (x) < (, x (a, b) 5 (a, b) x f n (x) M n M n < 5

f n (x) < f n (x) < (, x (a, b) 6 (i)f n (x), n = 1,..., (a, b) f n (x) f(x), x (a, b) f(x) (a, b) (ii)f n (x), n = 1,..., (a, b) f n(x) < (, x (a, b) f(x) = f n(x) (a, b) f n (x) = 1, n = 1, 2,.. (x + 1) n (i) f(x) = {, < x 1 1, x = (ii) (, 1) (iii) [1, 2] (i) (ii) 1 (x+1) n < ε n > log 1 ε log(1 + x) x lim x + log 1 ε log(1 + x) = 6

x n (iii)1 x 2 2 (x + 1) 3 1 (x+1) n 2 n (x + 1) n 3 n 1 2 n 1 (x + 1) n 1 3 n ε n 1 2 n ε ε x (1 x 2) lim f n (x) = 1.3 1.3.1 f n(x) f n (x) f n (x) = c n x n c nx n x = 7 n= c nx n x x x x n= c nx n n= c nx n n= c nx n x R x < R x n= c nx n x > R x n= c nx n R 7

8 f(x) = n= c nx n c ρ = lim n+1 c n R R = 1 ρ ( ρ = lim cn 9 f(x) = n= c nx n f (p) (x) = n(n 1)...(n p + 1)c n x n p, (p = 1, 2,...)...( ) n=p c p = f (p) (), (p = 1, 2,...) p! ( 1. ( ) 2. n= c nx n n= c n(x a) n a 1.3.2 1 [a, b] y = f(x) (n 1) (n 1) (a, b) n f(b) = f(a)+ (b a) f (a)+...+ 1! (b a)k f (k) (a)+...+ k! 8 (b a)n 1 f (n 1) (a)+r n (n 1)!

R n R n = (b a)n f (n) (c), a < c < b n! b = x f(x) = f(a) + R n = (x a) f (a) +... + 1! (x a)n f (n) (c), a < c < x n! a = (x a)k f (k) (a) +... k! (x a)n 1 f (n 1) (a) + R n, (n 1)! f(x) = f() + x 1! f () +... + xk k! f (k) () +... + xn 1 (n 1)! f (n 1) () + R n, R n = xn n! f (n) (c), < c < x ( R n c c = a + θ(b a) = (1 θ)a + θb, c = a + θ (x a) = (1 θ )a + θ x, c = θ x, < θ, θ, θ < 1 11 R n lim R n = f(x) = f(a) + (x a) f (a) +... + 1! (x a)n f n (a) +... n! f(x) = f() + x 1! f () +... + xn n! f n () +... 9

12 x lim x n n! = f (n) (c) f(x) = 1 1 x x < 1 n= xn x < 1 f(x) = 1 1 x x < 1 1 1 x = 1 + x + x2 +... + x n +... f(x) = e x f(x) = e x f (x) = e x, f (k) (x) = e x, f (k) () = 1, k =, 1,... f (n) (c) = e c e c M M x x N x x N x x lim R n = e x = 1 + x 1! +... + xn n! +... f(x) = sin x f(x) = sin x f (x) = cos x = sin(x + π 2 ), f (k) (x) = sin(x + kπ 2 ), k =, 1,... 1

f (2k+1) () = ( 1) k, f (2k) () =, k =, 1,... f (n) (c) = sin(c + nπ 2 ) x x f (n) (c) sin(c + nπ 2 ) 1 lim R n = sin x = x x3 3! + x5 5!... + ( 1)k x 2k+1 (2k + 1)! +... f(x) = cos x f(x) = cos x = sin(x + π 2 ) f (k) (x) = sin(x + kπ 2 + π ), k =, 1,... 2 f (2k) () = ( 1) k, f (2k+1) () =, k =, 1,... f (n) (c) = sin(c + nπ 2 + π 2 ) f (n) (c) 1 x lim R n = x cos x = 1 x2 2! + x4 x2k... + ( 1)k 4! (2k)! +... 11

f(x) = log(1 + x) f(x) = log(x + 1) f (x) = 1 = (x + 1) 1 (x + 1) f (k) k 1 (k 1)! (x) = ( 1) (x + 1), f (k) () = ( 1) k 1 (k 1)!, k = 1,... k R n = xn n! f (n) n 1 xn (c) = ( 1) n ( 1 1 + c )n, c x 1 < x 1 1 < x 1 lim R n = log(x + 1) = x x2 2 + x3 xn... + ( 1)n 1 3 n +... (x + 1) m f(x) = (x + 1) m m m (x + 1) m 2 m f(x) (m + 1) f(x) = (x + 1) m 1 < x 1 (x+1) m = 1+ m 1! 1) x+m(m x 2 m(m 1)(m 2)...(m n + 1) +...+ x n +... 2! n! 1. x n n=, 2. n! n= n!xn, 3. x n n=, 4. n 2 n= 12 x n n 3 +1,

5. n= (n!) 2 x n 6. xn (2n)! n= ( 1)n 7. n! n= xn2 1. cosh x = 1 2 (ex + e x )2. log(x + 1 + x 2 )3. log(1 x + x 2 )4. sin 3 x, 5.a x 6. log(1 x)7. sin 2 x8. log( 1+x 1 x ) 2 2.1 f(x) x T > f(x + T ) = f(x) f(x) T x n n= c nx n (T = 2π 13 [, π] f(x) f(x) = (a n cos nx + b n sin nx) n= a n, b n a n = 1 π b n = 1 π a = 1 2π f(x) cos nxdx, n = 1, 2,... f(x) sin nxdx, n =, 1, 2,... f(x)dx 13

sin mx cos nxdx =, m, n =, 1,... cos mx cos nxdx =, m n sin mx sin nxdx =, m n cos 2 nxdx = π, n = 1,... sin 2 nxdx = π, n = 1,... 2.2 f(x) 2π [, π] f(x), [, π] f(x) {x i } n i=1 x i f(x i +) = lim x xi + f(x) f(x i ) = lim x xi f(x) f(x i + ) = f(x i ) x i f(x) [, π] f(x) a n, b n a n = 1 π b n = 1 π a = 1 2π f(x) cos nxdx, n = 1,... f(x) sin nxdx, n =, 1,... f(x)dx (a n cos nx + b n sin nx) n= f(x) (a n cos nx + b n sin nx) n= 14

14 [, π] f(x) a n, b n n M a n, b n M 15 ( [, π] f(x) a n, b n a n, b n, (n ) f(x) f(x) f (x) f(x) resp. f(x) = f( x) resp.f(x) = f( x)) n b n = (resp.a n = ) resp. (resp. { { 1, x π x, x π f(x) = f(x) =, x 1, x sin x, x π f(x) = x, ( x π) f(x) =, x f(x) = x 2, ( x π) f(x) = sin x, ( x π) 2 [, π] f(x) [, π] 2π [, π] (i) π [, ] f(x) = f(x + π), x { (ii) [, ] f(x) = f( x), x 15

(iii) [, ] f(x) = f( x), x f(x) = x( x π) π [, ] a n = 2 π x cos 2nxdx = 2 π {[x 1 sin 2nx]x=π x= 1 2n 2n sin 2nxdx} = 1 [cos 2n 2 2nx]x=π x= =, (n ) π a = 1 π b n = 2 π xdx = π 2, x sin 2nxdx = 2 π { [x 1 cos 2nx]x=π x= + 1 cos 2nxdx} 2n 2n = 2 π { π 1 cos 2nπ + [sin 2nx]x=π 2n 4n2 x=} = 1, (n ) n π 2 1 sin 2nx n [, ] b n =, a n = 2 π 16 x cos nxdx

= 2 π {[x 1 sin nx]x=π x= 1 n n sin nxdx} = 2 [cos n 2 nx]x=π x= = 2 π n 2 π (( 1)n 1), (n ), a = 1 π xdx = π 2 π 2 + 2 n 2 π (( 1)n 1) cos nx [, ] a n =, = 2 π { [x 1 n b n = 2 π x sin nxdx cos nx]x=π x= + 1 n cos nxdx} = 2 π { n cos nπ + 1 [sin nx]x=π n2 x=} = 2 n ( 1)n+1, (n ) 2 n ( 1)n+1 sin nx 16 N A n, B n N (A n cos nx + B n sin nx) 17

,. (f(x) N (A n cos nx + B n sin nx)) 2 dx A n = a n, B n = b n 17 D n (x) = 1 + cos x + cos 2x +... + cos nx 2 (i)d n (x) = sin(n + 1)x 2 2 sin x x = 2 (ii) 1 π D n (x)dx = 1 (iii) f(x) [, π] 2π 1 π lim f(y)d n (x y)dy = 1 {f(x + ) + f(x )} π 2 f(x) 1 lim π f(y)d n (x y)dy = f(x) ( {D n (x)} n δ(x) sin(n+ D n (x) (i) D n () = lim 1 2 )x x = n + 1 2 sin x 2 2 (1)δ() = lim D n () = lim (n + 1 2 ) = 18

x lim D n (x) (ii) (2) δ(x)dx = π D n (x) (iii) (3) 1 π f(y)δ(x y)dy = f(x) δ(x) π (iii) 18 f(x) [, π] 2π 1 2 {f(x + ) + f(x )} = (a n cos nx + b n sin nx) n= f(x) f(x) = (a n cos nx + b n sin nx) n= ( f(x) n= (a n cos nx + b n sin nx) f(x) 19 1 π f(x) 2 dx a2 π 2 + (a 2 n + b 2 n) ( f(x) ( a 2, 1 2 (a2 n + b 2 n) 19