Limites de suites, cours, première S - MathsFG

(un)
(un)ll
l(un)
llimn→+∞un=l
(un)l
(un)l l0l
l0< l l0> l e
l0< l −e(un)l N
]l−e;l+e[
]l0−e;l−e[l0
N
l0
(un) (un)p un+1 =f(un)n≥p
l n
e l
p upl e
u up
n p
|u−l|< e
u f(u)
n n + 1
n
(un)un+1 =u2
nn u1= 0,75 n p
(un)
l= 0 e er e

P U E L
ABS(U−L)>
E
P+ 1 BP
U2BU
P
P→P
U→U
L→L
E→E
ABS(U−L)>
E
P+ 1 →P
U2→U
P P
(1
n)1
√n)1
npp
•limn→+∞
1
n= 0
•limn→+∞
1
√n= 0
•limn→+∞
1
np= 0
e
1
N< e (1 n)n > N 1 n< e 1 n
(un) (vn)l l0
•(un+vn)l+l0
•(un×vn)ll0
•k(kun)kl
•l06= 0 (un
vn)l l0
u v w v w
l vn≤un≤wnu
l

l e
]l−e;l+e[ (vn)l
N vnl−e(wn)N0
wn< l +e N00 vn≤un≤wn
N N0N00 un≤vn> l −e un≥wn≤l+e wn
l
(un) +∞+∞
−∞ [a; +∞[a
]−∞;a]
limn→+∞un= +∞limn→+∞un=−∞
•limn→+∞n= +∞
•limn→+∞√n= +∞
•limn→+∞np= +∞
p
]a; +∞[n n > a n
]a; +∞[ +∞n n > a2n2> a
(n2) +∞
(un) +∞(vn)
vn≤un(vn) +∞
]a; +∞[a(un) +∞
N(un) ]a; +∞[
N0vn≤unN N0(vn) ]a; +∞[
(vn) +∞

q
•q > 1 (qn) +∞
• −1< q < 1 (qn)
•q≤ −1 (qn)