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En raison de limitations techniques, la typographie souhaitable du titre, «
Série numérique : Rappels Série numérique/Rappels », n'a pu être restituée correctement ci-dessus.
Définition
∑
i
=
m
n
a
i
{\displaystyle \sum _{i=m}^{n}a_{i}}
a
m
+
a
m
+
1
+
.
.
.
+
a
n
−
1
+
a
n
{\displaystyle a_{m}+a_{m+1}+...+a_{n-1}+a_{n}}
modifier
∑
i
=
m
n
(
a
i
+
b
i
)
=
∑
i
=
m
n
a
i
+
∑
i
=
m
n
b
i
{\displaystyle \sum _{i=m}^{n}(a_{i}+b_{i})=\sum _{i=m}^{n}a_{i}+\sum _{i=m}^{n}b_{i}}
∑
i
=
m
n
λ
(
a
i
)
=
λ
(
∑
i
=
m
n
a
i
)
{\displaystyle \sum _{i=m}^{n}\lambda (a_{i})=\lambda \left(\sum _{i=m}^{n}a_{i}\right)}
(
∑
i
=
m
n
a
i
)
(
∑
j
=
p
q
b
j
)
=
∑
i
=
m
n
(
∑
j
=
p
q
(
a
i
b
j
)
)
=
∑
j
=
p
q
(
∑
i
=
m
n
(
a
i
b
j
)
)
{\displaystyle \left(\sum _{i=m}^{n}a_{i}\right)\left(\sum _{j=p}^{q}b_{j}\right)=\sum _{i=m}^{n}\left(\sum _{j=p}^{q}(a_{i}b_{j})\right)=\sum _{j=p}^{q}\left(\sum _{i=m}^{n}(a_{i}b_{j})\right)}
∑
i
=
m
p
a
i
=
∑
i
=
m
n
−
1
a
i
+
∑
i
=
n
p
a
i
=
∑
i
=
m
n
a
i
+
∑
i
=
n
+
1
p
a
i
{\displaystyle \sum _{i=m}^{p}a_{i}=\sum _{i=m}^{n-1}a_{i}+\sum _{i=n}^{p}a_{i}=\sum _{i=m}^{n}a_{i}+\sum _{i=n+1}^{p}a_{i}}
m
≤
n
≤
p
{\displaystyle m\leq n\leq p}
On a donc :
∑
i
=
m
n
a
i
+
∑
i
=
n
p
a
i
=
∑
i
=
m
p
a
i
+
a
n
≠
∑
i
=
m
p
a
i
{\displaystyle \sum _{i=m}^{n}a_{i}+\sum _{i=n}^{p}a_{i}=\sum _{i=m}^{p}a_{i}+a_{n}\neq \sum _{i=m}^{p}a_{i}}
(
∑
i
=
1
n
a
i
b
i
)
2
≤
(
∑
i
=
1
n
(
a
i
)
2
)
(
∑
i
=
1
n
(
b
i
)
2
)
{\displaystyle \left(\sum _{i=1}^{n}a_{i}b_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}(a_{i})^{2}\right)\left(\sum _{i=1}^{n}(b_{i})^{2}\right)}
