1 x(x + 1)
n1
1 n(n + 1)
n n(n + 2) (n + 1)2
n1
un = cos yn = n( n(vn )
(sin θ)
a
n
n
n0
e1/n
n0
3
n
(a
∈ R
2n
θ
∈ R)
π 2
2n + 3n + n4 n5n + 7n 7n7 + 2 1 1+ 1 n
− 1 − n1
√ n xn
sin nθ
einθ θ =
−
vn =
xn = n( n(un )
n 0
2n3 1 n4 + n + n + 1
1 n
(n n( n(n))2 + n n2 + n4 n( n(n) 1/n
√ −
0
nα u
n
u
n
n 0
un
√ un
un
n(1 n(1 + u + un )
sin2 t dt
π sin
1 2n
2n n!
n2 3n
nn n!
√ e−n+ n
n n + 1
n2
n!
∼
n √ 2πn n
e
1/x 1 1 + + 2 3
· ·· + n1 n n 1 + 12 + 13 + · ·· + n −1 1 .
1 1 S n = 1 + + + 2 3 1/x 1 n + 1
· · · + n1 − n n
n + 1
n
n
1 . n
S n
( 1)n 2n + 7
γ 0, 577
∼
( 1)n n2 + 2n + 3
−
−
−
n
un =
u
(−1) √ n + ( −1)n
vn =
einθ n
n(n) n
( 1)n
cos nθ n3 + 1
√
θ
∈ R
( 1)n , n2 n
−√
∀
n
n2
”
u
n
v u ∼ v , n
n
u
lim un = 0
n
+
∞
u
n
n
u
n
un
√
−
∞
un vn
( 1)n n + ( 1)n
+
u
( n + ( 1)n+1 )
n
u2n + u2n+1 n2 + 2 sin n n9 + 5n7 + 7
√
n
tan
(−1) n!
+ nπ
−
−
cos nθ n 1 + n
n(1+ x)
θ
∈ R.
u v n
n
n0
1 un = v n = n , n 2
n0
∀ ∈ N n+1
(−1)
n1
1
n2
1 1 1 1 1 − 212 + 312 − 412 + 512 + 712 − 612 · · · − (2k) ·· · − ·· · + + + 2 (2k + 1)2 (2k + 3)2 (2k+1 − 1)2 (2(k + 1)) 2
log(n) √ n 1 1 + 1 2 n 2n + 3 n1
n2
n
n1
n
n0
n!
(−1) sin(n) n 1 − n 1 + 1 n n e n
2
n 1
n1
in
n0
π 11
n + 1
n
(−1) n−1 sin 1 + 1 n (−1) sin 1
n2
n1
n
n1
n
2n +√ 5 (n + 1)( n + 2) (−1) n(n) n + n n 2
n 0
n
2
n 1
n(n)
n2
(n n)n
n
2 n
n0
n0
n
n1
nn
n2
1 zn 2 n +1
1 (x n 3n
n 0
n 1
sin 1 x n 2x
(−2) x n! n x
nx 1x
n
n2
− 2)
n0
αn n z n n
∈ C
n0
n
n 1
a0 , a1 , ... , a n , ... , z 0 , z
n
2
n0
n
n0
n
n
n0
(n + 2)2
n0 +
∞ z
n=0
(n + n + 1)x 2
n0
n
+
∞
n
n!
n=0
n
(−1)
n1
n
+
+ 4n
n=0
n
(n + 1)!
zn
n3 x
n 2n
xn
n 1
f (x) = (1 + x)−2 (1 + x)−1 (1 + x)−1 (1 + x)α
α =
−2
n
(n + 1)!
∞ (n + 2)2
zn (n + 1)!
(α
∈ R)
n2 z
− 1)n (z ∈ C)
a z a z
|z| > |z0|
n0
(z
3n+1
nα
∈ C
|z| |z0|
n n 0
n
n1
n
1 n z n!
α
∈ R)
n0
n1
n 2n
(x
n n 0
n
n1
n
e
a z a z
n(n) x n α x
n
2
n n
2n
n
zn
n
2n
f n
1 2 + 3x
n
1 + x 1
−x
√ 1 1− x2
x
x
e2x (1 x2 )y xy = 0 () ()
−
()
1 cos 10
cos x 10−12
−
f
R
f (x) = 1
R
x
2π f
∈]0, π[ , f
f +
∞
p=0 +
f 1 , f 2
f 3
R
p=0
| | ∀ x ∈ [−1, 1[
∀ ∈ −
f 1 , f 2
( 1) p , 2 p + 1
−
+
∞
+
∞
1 , n2
n=1
, f 3 (x) =
f 3
p=0
0 , 2x ,
f 1
f 1 , f 2
∀ x ∈ [−1, 0] ∀ x ∈ ]0, 1[
.
f 2
f 3 +
∞
1 , (2 p + 1)2 f
p=0
R
+
∞
1 (2 p + 1)4
n=1
1 n4
f (x) = x(1
R
[0, 1] f
f f +
∞
n=1
f
+
∞
1 n2
1 n4
n=1
n1 +
∞
p=0
( 1) (2 p + 1)3
−
g1 (x) = cos x , x
| ∀ ∈ R
+
n=1
g2
R
0 x g (x) = cos − cos x
, , ,
2
g1
∞
f sin2πnx π3 n3
p
g1
|
1 (2 p + 1)2
R
f 1 (x) = x , x [ 1, 1[ , f 2 (x) = x ,
p=0
+
∞
n2
n=1
+
−
∞ 1
∞
( 1) p 2 p + 1
( 1)n , 4n2 1
−
−
R
x = kπ (k Z) x ]2k π , (2k + 1)π[ (k Z) x ](2k + 1) π , (2k + 2)π[ (k
∈ ∈
g2
+
∞
n=1
1 4n2
−1
+
,
∞
n=1
1 (4n2
− 1)2
∈
∈
∈ Z)
− x) , ∀x ∈
x = y 2 0 y 1; x = a cos t y = a sin t z = bt ρ = a(1 + cos θ) a > 0 0
≤ ≤
0 ρ
≤ t ≤ t0; ≤ ≤ a
(x + y) dx + (x
C
C
x = cos t
− y) dy 0
2π.
0
2π.
A = (1, 1, 1) B = (2, 2, 2). x = cos t y = sin t z = t t 0
2π.
y = sin t t
xydx + (x + y) dy
C
C
x = cos t
y = sin t t
(y + z) dx + (z + x) dy + (x + y) dz x2 + y 2
C
C C
y 2 dx
Γ
Γ Γ
− x2 dy
A = (1, 0) B = (0, 1). (0, 0) 1
xy dx + x y dy A = (1, 0) Γ
2
y =4
Γ = Γ1
− 3x
A O Γ3
B
∪ Γ 2 ∪ Γ 3 y = x
xy2 dx + x2 y dy u(x, y)
2
B = (0, 1).
2
Γ1 = AB B x O A xy2 dx + x2 y dy Γ
≥ 0
i
du =
Γ
Γ2 i = 1, 2, 3.
(x + 2y) dx + y dy (x + y)2
ω = P (x, y) dx + Q(x, y) dy P (x, y) = u
(3x2
− y2)(x2 + y2) x2 y
D
Q(x, y) =
D = (x, y); x > 0, y > 0 du = ω
{
}
ω
(3y2
− x2)(x2 + y2) . xy 2
ω
0
≤ t ≤ 2π.
x = t + cos2 t y = 1+ sin2 t
Γ
Γ
−→ V ( x, y) = (y 2 , x2 )
x2 +4y2 4 = 0; y
−
(x, y) = (cos(x), sin(y)) V
−→ V ( x, y) = (x2 + y 2 , x2 − y 2 )
A = (1, 0) B = (0, 1)
−→ −→ −→ −→ F = (y + z) i + (x + z) j + (x + y) k O
C = (1, 1, 1),
(OC ). x = t, y = t 2 , z = t 3 .
−→ −→ −→ −→ G = (x + yz) i + (y + xz) j + (z + xy) k .
≥ 0
4
12x
(
f (x, y) dy) dx
3x2
0
1
3x
(
0
f (x, y) dy) dx
2x
√
2ax x2
a
−
(
a 2
f (x, y) dy) dx
0
D y 2 = x.
y = x,
a)
(x
− y) dxdy
D
x = 0,
xy dxdy
D
y = x + 2,
y =
−x
D
D
y = x2 ,
[0, 1]
×
f : [0, 1] [0, 1]
× [0, 1] → R
y = x 3 . y = x
T 1
T 2 .
f (x, y) dxdy =
T 1
f (x, y) dxdy.
T 2
u = y, v = x,
T 1
D D = (x, y)
{
I =
| 0 ≤ x, 0 ≤ y,
D
(4
x2 + y 2
− x2 − y2) dxdy.
≤ 1}
xy dxdy =
T 2
xy dxdy.
ρ = 1 + cos θ. I = I
√ xy e− − dxdy x y
T a
T = {(x, y) ∈ x = tu a
y = (1
R2 /x
≥ 0; y ≥ 0; x + y ≤ a}
− t)u
D
r = p(θ)
0
≤ θ ≤ 2π
θ = 0
≤ r ≤ p(2π)
p(0)
1 2π 2 (D) = p (θ) dθ. 2 0 r = a(1 + cos(θ))., b) r = te φ = te te x = y = te
D
A
a)
C C
r = aθ x,y. r, φ
C C
x2 + y 2
D 2
I y = x
2
x = y
I =
(2xy
γ
n
lim I n ,
−→+∞
n
∈ N,
I n =
2
γ
− x2)dx + (x + y2)dy.
2
e−x dx. Dn = x2 + y 2
{ ≤ x ≤ n, 0 ≤ y ≤ n}.
{
C n = 0
(x2 +y 2 )
e− e− K =
J n = n
Dn
C n
Dn , C n
D2n
→ +∞
J n
n
− 2y = 0
D (x − y )dxdy
I =
0
n
a > 0
2
dxdy
(x2 +y 2 )
+ 0
∞ e−
x2
dx,
≤ n2, x ≥ 0, y ≥ 0} J 2n =
D2n
e−(x
K n = I n2 .
dxdy. J n
J 2n ?
a > 0
K n ?
≤ K n ≤ J 2n. I.
2
+y 2 )
dxdy
(O, i, j, k)
1 4 7 2 , A = B = 5 , C = 8 . 3
6
9
B, B A
A.( A ∧ B), A , A + B , (A ∧ B) ∧ C, A ∧ (B ∧ C ). ∧ A, A. B,
∧
(O, i, j, k)
R3
√ 0 0 , v = 0 , w = u = √ −1 . 1 2
3 2
3 2
− 12
0
(u, v, w) u, v
w
u.v , u.w, v.w
−→ −→ −→
R3
(O, i , j , k )
u v w u = u , v = v , w = w . 1
u
A
1
2
2
2
u3
v3
w3
∧ (v ∧ w) = (u.w) v − (u.v )w. E = R2 M ∈ E M ∈ E
B
[AB ]
1
= 0 AM.AB = 0 AM.BM
I
E = R3 ? θ P
P 1 (1, 2, 3)
−→ → −b = b1−→i + b2−→ −→ −→a = a1−→i + a2−→ j + a3 k j + b3 k c = det
c = ab sin θ, U
a a 2
3
b2
b3
i + det
a a 1
3
b1
b3
j + det
a a 1
2
b1
b2
θ
V
−→ −→ −→ −→ −→ −→ U + V 2 + U − V 2 = 2( U 2 + V 2 ).
k
a
b.
P 2 (2, 3, 1).
− −
U
V
−→ −→ −→ −→ U + V 2 = U 2 + V 2 . AM AN A = (2, 1, 1) M = (3, 0, 2) N = (4, 2, 1)
−
−
−
R3
∈ R3 M ∈ R3 M
U
||U − V 1||≥ ||U ||−|| V ||. 2 || || − ||U − V ||2 = + V U.V 4 U
V
AP
P = (5, 3, 0).
−→ −→ −→
(O, i , j , k ).
−−→ ∧ −AB = −→ −→0 −−→ −−→ −→ AM ∧ BM = 0 AM
R3
