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I R (f n )n
f n : I −→ Rx −→ f n (x) ∀n∈IN.
f n f f n+ ∞
−→ f f I
I = [0, 1] f n (x) = xn ∀x∈I ∀n∈ IN
(f n )n I f n (x) n x
(f n )n I (f n )n f I x∈I
(f n (x))n f (x)
∀x∈I, ∀ε > 0, ∃n ε,x ∈IN / ∀n∈IN, n ≥n ε,x =⇒ |f n (x) −f
(x)| ≤ε f (f n )n I
f nCV S
−→f I.
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I = [0, 1] f n (x) = xn x
∈
[0, 1]
x∈[0, 1[ xn −→0
x = 1 1n = 1 ∀n∈IN
f nCV S
−→f I
f (x) = 0 x∈[0, 1[1 x = 1.
f n [0, 1] f [0, 1[
I = [−1, 1] f n (x) = xn
n x∈I
xn
n ≤ 1n ∀x∈I, ∀n∈IN
∗.
limn → + ∞
f n (x) = 0 f nCV S
−→0 I I = R f n (x) =
xn
x R f nCV S
−→0 R
I =]0, 1] f n (x) =
1x
1n ≤x ≤1
n 0 < x ≤ 1n
f nCV S
−→ f I f (x) = 1 /x x ∈]0, 1] n0 x ≥
1n ∀n ≥n0 =⇒f n (x) = 1 /x, ∀n ≥n0
(f n )n I (f n )n f I
supx∈I |f n (x) −f (x)| −→ 0 n +∞
∀ε > 0, ∃nε ∈IN / ∀n∈IN, ∀x∈I, n ≥n ε =⇒ |f n (x) −f (x)| ≤ε
f n
CV U
−→f I
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(f n )n (gn )n f g I (f n gn )n fg I
α β ∈R f n
CV U
−→ f I gnCV U
−→ g I
supx∈I |(αf n (x) + βgn (x)) −(αf (x) + βg(x))| ≤ |α|supx∈I |f n
(x) −f (x)|
+ |β |supx∈I |gn (x) −g(x)|.
f nCV U
−→f I =⇒supx∈I |f n (x) −f (x)| −→0 n −→+ ∞gn
CV U
−→g I =⇒supx∈I |gn (x) −g(x)| −→0 n −→+ ∞.
f nCV S
−→f I gnCV S
−→g I x∈I lim
n → + ∞(αf n (x) + βgn (x)) = α lim
n → + ∞f n (x) + β lim
n → + ∞gn (x) = αf (x) + βg(x).
f nCV S
−→f I gnCV S
−→g I x∈I limn → + ∞ (f n (x) gn (x)) = limn → + ∞ f n (x) limn
→ + ∞ gn (x) = f (x) g(x).
I = R f n (x) = x + 1n
x ∈ I limn → + ∞ f n (x) = x f n
CV S
−→f I f (x) = x supx∈R |f n (x) −f (x)| =
1n −→0 n −→+ ∞,
f nCV U
−→f I (gn )n gn = f n ×f n = f 2n
gnCV S
−→ g = f 2 I |gn (n) −g(n)| = 2 +
1n2 ≥2, ∀n∈IN
∗ supx∈I |gn (x) −g(x)| > 2
(gn )n g
(f n )n I (f n )n f I (un )n
I limn → + ∞ |f n (un ) −f (un )| = 0
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f nCV U
−→f I (un )n I |f n (un ) −f (un )| ≤supx∈I |f n (x) −f (x)| −→0
n −→+ ∞.
limn → + ∞ |f n (un ) −f (un )| = 0 , ∀(un )n I .
(f n )n f I
∃ε > 0, ∀n ε ∈IN / ∃nk > n ε supx∈I |f n (x) −f (x)| >
ε > ε − 1k
.
sup
(un k )n k
I
|f n k (un k ) −f (un k )| ≥ε − 1k
.
k +∞ ε ≤0
I = [0, 1] f n (x) = xn f nCV S
−→f I
f (x) = 0 x∈[0, 1[1 x = 1.
(f n )n
f
I
(un )n
un = 1 − 1n
.
un ∈I, ∀n∈IN∗ f n (un ) = 1 −
1n
n
−→ 1e
f (un ) = 0 , ∀n∈IN∗,
|f n (un ) −f (un )| −→ 1e = 0 .
f n f I
(f n )n I (f n )n lim
n → + ∞supx∈I |f n + p(x) −
f n (x)| = 0 , ∀ p∈IN ∀ε > 0, ∃n ε ∈IN / ∀(n, p)∈IN2, n ≥n ε
=⇒supx∈I |f n + p(x) −f n (x)| ≤ε.
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(f n )n I f n CV U −→f I
f n I =⇒ f I
f n I =⇒ f I
f n I n M n > 0
|f n (x)| ≤M n , ∀x∈I. x∈I
|f (x)| ≤ |f (x) −f n (x)|+ |f n (x)| ≤supx∈I |f n (x) −f (x)|+
M n , ∀n∈IN n∈IN
n = 0
|f (x)| ≤supx∈I |f 0(x) −f (x)|+ M 0, ∀x∈I. f I
x0 ∈I f x0 n∈IN |f (x) −f (x0)| ≤ |f (x) −f n (x)|+ |f n (x) −f
n (x0)|+ |f n (x0) −f (x0)|
≤ 2supx∈I |f n (x) −f (x)|+ |f n (x) −f n (x0)|.
ε > 0
f nCV U
−→f I =⇒ limn → + ∞ supx∈I |f n (x) −f (x)| = 0=⇒ ∃n0 ∈IN, ∀n
≥n0 ⇒supx∈I |f n (x) −f (x)| ≤
ε4
.
n = n0 supx∈I |f n 0 (x)−f (x)| ≤
ε4
n = n0
|f (x) −f (x0)| ≤ ε2 + |f n 0 (x) −f n 0 (x0)|.
f n 0 I x0
∃η > 0, ∀x∈I / |x −x0| ≤η ⇒ |f n 0 (x) −f n 0 (x0)| ≤ ε2
.
∃η > 0, ∀x∈I / |x −x0| ≤η ⇒ |f (x) −f (x0)| ≤ε. f x0
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f n (x) = sin nx
n
R f n (x) = cos nx, ∀n ∈ IN∗
f n (x) = xn
f (x) = 0 x∈[0, 1[1 x = 1.
f [0, 1] f n [0, 1] f n f [0, 1]
(f n )n I = [a, b] f I [α, β ]⊂[a, b]
limn → + ∞
β
αf n (x) dx =
β
αlim
n → + ∞f n (x) dx =
β
αf (x) dx.
n
β
αf n (x) dx −
β
αf (x) dx =
β
α(f n (x) −f (x)) dx
≤ β
α |f n (x) −f (x)|dx ≤ (β −α)supx∈I |f n (x) −f (x)|.
limn → + ∞
β
αf n (x) dx =
β
αf (x) dx.
(f n )n I = [a, b] f I x∈I
F n (x) = x
af n (t) dt F (x) =
x
af (t) dt.
F nCV U
−→F I. (f n )n C 1
I = [a, b]
f nCV S
−→f I. f n
CV S
−→g I.
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f nCV U
−→f I f I f = g
f n (x) = f n (a) + x
a f n (t) dt
I = [−1, 1] f n (x) = x2 + 1 /n f n I f n f n f I f (x) = |x| f
n f I
x2 + 1n − |x| = x2 + 1n −√ x2 = 1/n x2 + 1 /n + √ x2 ≤ 1/n1/ √ n
=
1√ n −→0
f (x) =
|x
| 0
I R (f n )n R
n
S n =
n
k=0f k .
f n (S n )n
n∈IN
f n n ≥ 0
f n
n f n n f n (x) = an xn an ∈ R x ∈ R
n∈IN
an xn
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n∈IN
f n
I I
n∈IN
f n S I
x∈I
n∈IN
f n (x) S (x)
n∈IN
f n x∈I
n∈IN
f n (x)
n∈IN
xn S ]−1, 1[
S (x) = 11 −x
x ]−1, 1[
S n (x) =n
k=0
xk = xn +1
1 −x+ ∞
−→+ ∞
n =0
xn = S (x) = 11 −x
.
|x| ≥1 n∈IN
xn 0
n +∞
n∈IN
f n I
(f n )n 0 I
x ∈ I n∈IN
f n I
n∈IN
f n (x)
limn → + ∞
f n (x) = 0
n∈IN
xn |x| ≥ 1 x ∈]−∞, −1]∪[1, + ∞[
limn → + ∞
xn = 0 .
n∈IN
f n
S I S I
limn → + ∞
supx∈I |S n (x) −S (x)| = 0 .
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n∈IN xn
S (x) = 11 −x
[−r, r ]
|r | < 1
|S n (x) −S (x)| =1 −xn +1
1 −x − 11 −x
= |x|n +11 −x ≤
rn +1
1 −r, ∀x∈[−r, r ].
supx∈[− r,r ] |S n (x) −S (x)| ≤
rn +1
1 −r+ ∞
−→0, ∀ |r | < 1.
]−1, 1[
f n (x) = sin(nx )
n2 , ∀n∈IN∗, ∀x∈R .
n x
|f n (x)| ≤ 1n2
.
n∈IN
f n (x)
n∈IN
f n (x)
R
n∈IN
f n
I
∀ε > 0, ∃n ε ∈IN / ∀(n, p)∈IN, n ≥nε ⇒supx∈I n + p
k= n +1f k(x) ≤ε.
n∈IN
f n I
(S n )n
S n + p −S n =n + p
k= n +1
f k
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n∈IN
un f n
un ∈R , ∀n∈IN
(f n )n
I
(un )n 0
(n
k=0
f k)n I n∈IN
∃M > 0 / ∀x∈I, ∀n∈IN, |Bn (x)| =n
k=0
f k(x) ≤M.
n∈IN
un f n I
n ≥ 1
sin nxn
[r, π −r ] ∀r∈]0, π[
un = 1n
f n (x) = sin nx, ∀n∈IN∗
. (un )n
Bn (x) =n
k=1
sin kx = sinnx2
sin (n +1) x2sin(x/ 2)
|Bn (x)| ≤ 1sin( x/ 2) x ∈ [r, π −r ] x2 ∈ [r2 , π2 − r2 ] r∈]0,
π[
0 < sin(r/ 2) ≤sin(x/ 2) ≤1
|Bn (x)| ≤ 1
sin(r/ 2) , ∀x∈[r, π −r ].
n ≥ 1
sin nxn
n∈IN
f n
n∈IN|f n | I
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n ≥ 1
xn2
R
x R
n ≥ 1
xn2
α = 2
n∈IN
f n
(un )n |f n (x)| ≤un ∀x∈I ∀n∈IN
n∈IN
un
n ≥ 1
sin nxn2
R
sin nxn2 ≤
1n2
, ∀x∈R , ∀n∈IN∗
n ≥ 1
1n2
n∈IN
e− n2 x
[r, + ∞[ r > 0
e− n2 x ≤e− n
2 r , ∀x∈[r, + ∞[, ∀n∈IN∗. un = e− n
2 r un
limn → + ∞
un +1un
= limn → + ∞
e− (2n +1) r = 0 < 1.
n∈IN
un
n∈INe
− n 2 x
=⇒ =⇒
=⇒ =⇒
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n ≥ 1
(−1)nn + x
R +
x ∈ R + n∈IN
(−1)nn + x
|Rn (x)| =+ ∞
k= n +1
(−1)kk + x ≤
1n + 1 + x
.
x ≥0
supx∈R + |S n (x) −S (x)| ≤ 1n + 1 + ∞−→0 S (x) =
+ ∞
n =0(−1)
n
n + x .
(−1)nn + x
= 1n + x
n∈IN
(−1)nn + x
R +
(f n )n I
n∈IN
f n I + ∞
n =0
f n I
(f n )n I = [a, b]
n∈IN f n
I
+ ∞
n =0 f n
I
[α, β ]⊂[a, b]
+ ∞
n =0 β
αf n (x) dx =
β
α
+ ∞
n =0
f n (x) dx.
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(f n )n I = [a, b]
n∈IN
f n I x∈I
F n (x) = x
af n (t) dt.
n∈IN
F nCV U
−→F I
F (x) = x
a
+ ∞
n =0
f n (t) dt, ∀x∈I.
(f n )n
C 1
I = [a, b]
n∈IN
f n I
n∈IN
f n I
n∈IN
f n I C 1
+ ∞
n =0 f n (x) =
+ ∞
n =0 f n (x).
