TD N° 2 : Suites numériques - Exercices de convergence et limites

1
1.(un) (vn) (un+vn) (un−vn)
(un) (vn)
2.(un) (vn)
lim
n→+∞(un, vn).
3.(un) (vn)
0≤un≤1,0≤un≤1unvn→1.
2
1.
)un=3n−(−2)n
3n+ (−2)n)un=pn2+n−1−pn2−n+ 1
)un=n−√n2+ 1
n+√n2−1un=1
n2
n
X
k=1
k.
2.
a)un=1 + 1
nn
b)un=n
√n2c)un=sin 1
n
1
n
.
3.(un)
a)sin n
1+(−1)n+1 b)n!
nnc)n−(−1)n n+ (−1)n.
3
(un)l.
vn=u1+··· +un
n
(vn)
v2n≥un+vn
2.
vn→l

4
un=
n
X
k=1
1
√k−2√n vn=
n
X
k=1
1
√k−2√n+ 1.
(un) (vn)
n
X
k=1
1
√k.
5
1.(un) (u2n),(u2n+1) (u3n)
(un)
2.cos n
7
1.(un)
u0≥0∀n∈Nun+1 = 1 + (un).
2.(un)
u0∈R∀n∈Nun+1 = 1 + un.