Series2014 I
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Transcript of Series2014 I
7/25/2019 Series2014 I
http://slidepdf.com/reader/full/series2014-i 1/31
7/25/2019 Series2014 I
http://slidepdf.com/reader/full/series2014-i 2/31
+
+
+ · · · +
n + · · · =
( )
limn→+∞
+
+
+ · · · +
n
=
.
( ) ǫ > n
∈ N
+
+
+ · · · +
n −
< ǫ
n ≥ n
7/25/2019 Series2014 I
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7/25/2019 Series2014 I
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+∞
∑ n=
an. n ∈ N :
s n =n
∑ k
=
ak
{s n}
{s n}
S ∈ R,
n
→+∞
s n = S
+∞
∑
n=
an = S .
7/25/2019 Series2014 I
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a)+∞
∑ n=
(− )n C , C ∈ R.
b )+∞
∑ n=
(an+
− an, ) con {an} una sucesi on de n umeros reales
c )
+∞
∑ n=
n√ n −
n+ √ n+
.
d )+∞
∑ n=
C r n, r , C ∈ R.
e )+∞
∑ n=
[a + b (n − )] , a, b ∈ R.
f )+∞
∑ n=
n .
g )+∞
∑ n=
n! .
7/25/2019 Series2014 I
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+∞
∑ n=
an converge ⇔+∞
∑ n=n
an converge ,
n ∈ N
∑ an
n→+∞an =
Si
n→+∞
an
= , entonces ∑ an diverge
7/25/2019 Series2014 I
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a)+∞
∑ n=
n.
b )+∞
∑ n=
nr , r <
.
c )+∞
∑ n=
n+
n .
d )+∞
∑ n=
nπ
.
7/25/2019 Series2014 I
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∑ an ∑ b n α, β ∈ R,
∑ (αan + βb n)
∑ (αan + βb n) = α∑ an + β∑ b n.
∑ an
C = , ∑ C an
∑ an ∑ b n ∑ (an + b n)
∑ an ∑ b n α, β ∈ R,
∑ (αan + βb n).
7/25/2019 Series2014 I
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∑ an
an n ∈ N. ∑ an
∑ an ∑ b n c > n
∈ N
an ≤ cb n
n ≥ n
,
∑ b n ∑ an
∑ an ∑ b n
7/25/2019 Series2014 I
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a)+∞
∑ n=
nr , r > . b )+∞
∑ n=
n . c )+∞
∑ n=
n+
n−
+sen n
.
d )+∞
∑
n=
|cscn|n . e )
+∞
∑
n=
n+n
.
∑ an ∑ b n
n→+∞
an
b n= ∑ b n ∑ an
n→+∞
anb n
= C
C > ,
n→+∞
anb n
= +∞ ∑ b n ∑ an
7/25/2019 Series2014 I
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∑ an ⇔ (∀ǫ > ) (∃n
∈ N) ( n ≥ m ≥ n
⇒n
∑ k =m+
|ak | < ǫ )
∑ an
∑ |an|
∑ an ∑ |an|
∑ an
∑ r n
r ∈ (− , ),
+∞
∑ n=
(− )n+
n
7/25/2019 Series2014 I
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∑ an ∑ an
|∑ an| ≤ ∑ |an|
∑ b n b n ≥ , n ∈ N.
k >
n
∈ N
|an| ≤ k b n
n ≥ n
,
∑ an
n > n
, |an| ≤ kc n, < c < k >
∑ an
∑ an, α = limsup n |an|,
≤ α < ,
α >
,
α = ,
∑ an.
7/25/2019 Series2014 I
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+∞
∑ n=
nr .an r , a ∈ R.
+ a + a + a + · · · + a
n− + a
n + · · ·
7/25/2019 Series2014 I
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∑ an,
an =
,
n.
limsup |an+
||an|
< ,
liminf |an+
||an| >
,
limsup |an+
||an| =
liminf |an+
||an| =
,
∑ an.
7/25/2019 Series2014 I
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+∞
∑ n=
nr .an r , a ∈ R.
+ a + a + a + · · · + a
n− + a
n + · · ·
+∞
∑ n=
x n
n!
∑ an,
an =
n+
| n
n−
∤ n
+∞
∑ n=
n+(−
)n+
+∞
∑ n=
(−
)n
−n
a + ab + a b + a b + a b + · · ·
+∞
∑ n=
nn
n√ n!
7/25/2019 Series2014 I
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{an}
lim an+
an< lim n
√ an ≤ lim n
√ an < lim
an+
an
lim an+
an
lim n√ an
7/25/2019 Series2014 I
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f
x ≥ k , k ∈ N,
f (n) = an n ≥ k ,
∑ an,
+∞
k f (x )dx
p
q ,
+∞
∑ n=
n [lnn]p
+∞
∑ n=
[lnn]p
+∞∑ n=
nlnn [ln(lnn)]p
+∞
∑ n=
(lnn)q
np
7/25/2019 Series2014 I
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∑ an = A ∑ b n = B A, B ∈ R. n,
s n = ∑ nk =
ak t n = ∑ nk =
b k ,
AB = limn→+∞
(s n.t n)
= limn→+∞
(a
+ a
+ · · · + an)(b
+ b
+ · · · + b n)
= limn→+∞
a
b
+ a
b
+ · · · + a
b na
b
+ a
b
+ · · · + a
b na
b
+ a
b
+ · · · + a
b n+
· · ·+
anb
+ anb
+ · · · + anb n
∑ d n,
d
= a
b
d n = ∑ ni =
ai b n + ∑ n−
j =
an b j
n = , , , · · · ,
∑ d n = ∑ an . ∑ b n.
7/25/2019 Series2014 I
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+∞
∑ n=
an
+∞
∑ n=
b n, C k =
k
∑ n=
an b n−k , para k =
,
,
,
, · · ·
+∞
∑ k =
C k ,
+∞∑
n=
an
+∞∑
n=
b n,
+∞∑
n=
an b n
+∞
∑ n=
(−
)n+
n ,
7/25/2019 Series2014 I
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+∞
∑ n=
an
+∞
∑ n=
b n,
+∞
∑ n=
an
+∞
∑ n=
an = A
+∞
∑ n=
b n = B ,
+∞
∑ n=
an
+∞
∑ n=
b n A B .
∑ an ∑ b n
A
B ,
AB .
7/25/2019 Series2014 I
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+∞
∑ n=
an,
+∞
∑ n=
b n
+∞
∑ n=
c n
+∞
∑ n=
an = A,
+∞
∑ n=
b n = B
+∞
∑ n=
C n
+∞
∑ n=
an
+∞
∑ n=
b n,
+∞
∑ n=
C n = A B .
7/25/2019 Series2014 I
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+∞∑
n=
n(n+
)
n ∈ N,
an =
n , b n =
(n+
)
,
c n =
n(n+ )
d n =
(n+ )
,
+∞
∑ n=
n(n+ )
+∞∑ n=
an
+∞∑ n=
b n
+∞∑ n=
c n
+∞∑ n=
d n
{an}
{b n}
An =n
∑ k =
an
A− =
. p , q ∈ N ∪ { } ≤ p ≤ q ,
q
∑ n=p
anb n =q −
∑ n=p
an(b n − b n+
) + Aq b q − Ap −
b p
7/25/2019 Series2014 I
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+∞
∑ n=
an
+∞
∑ n=
b n
n
∑ k =
ak
+∞
n=
{b n}+∞n=
limn→+∞
b n = ,
+∞
∑ n=
anb n
{an}
limn
→+∞
an = , ∑ (− )n an
7/25/2019 Series2014 I
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+∞
∑ n=
n−
n +
+∞
∑ n=
(− )n nlnn[e ]n
+∞
∑ n=
(−
)n
n+
+∞
∑ n=
+αn , α >
− .
+∞
∑ n=
cos (nα)n
7/25/2019 Series2014 I
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φ : N →N, ∑ aφ(n)
∑ an
φ : N →N,
φ(n) =
n −
,
n
n + ,
n .
∑ +∞
φ(n) =
+
+
+
+ · · ·
φ : N →N,
φ(n) = (k − ) + (k − n) + (k − )
< k
,
∑ +∞
aφ(n) = a
+ a
+ a
+ a
+ a
+a
+ a
a
+ a
+ · · · an + an −
+ · · · a(n−
) +
+ · · ·
∑ +∞
an.
7/25/2019 Series2014 I
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φ : N →N, φ(
n + ) = n + , φ( n + ) = n + , φ( n + ) = n + ,
+∞∑
aφ(n) = +
−
+
+
−
+
−
+ · · ·
∑ +∞
(−
)n+
n .
−
−
+
−
−
+
−
−
+
− −+
− −+
· · ·
7/25/2019 Series2014 I
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+∞
∑
(− )n+
n =
−
+
−
+
−
+ · · · ,
+
−
+
+
−
+
+
−
+
· · ·
φ : N →N,
φ(n) =
k −
,
n =
k −
k −
,
n =
k −
k =
,
,
, · · ·
k ,
n =
k .
∑ +∞
(− )n+
n = ∑ +∞
an, ∑ +∞
aφ(n)
∑ +∞
(−
)n+
n = S ∑
+∞
aφ(n) = S ′,
S = S ′.
7/25/2019 Series2014 I
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∑ an
n ∈N,
p n = |
an
|+ an
= max {
an ,
}
q n = |
an
| −an
= max { −
an ,
}
an = , p n = q n = .
p n, q n ≥
,
n.
an = p n − q n
|an| = p n + q n,
n.
≤ p n ≤ |an| ≤ q n ≤ |an|
n.
7/25/2019 Series2014 I
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∑ p n ∑ q n
∑ an,
∑ an
n
∑ k =
ak =n
∑ k =
p n +n
∑ k =
q k
s n, P n Q n ∑ an, ∑ p n ∑ q n,
s n = P n − Q n,
n.
An
∑ |an|,
An = P n + Q n,
n.
∑ an ∑ p n ∑ q n
∑ an ∑ p n ∑ q n
7/25/2019 Series2014 I
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∑ an α, β ∈ R
α ≤ β. −∞ ≤ α ≤ β ≤ +∞.
∑ aφ(n)
s ′n,
lim s ′n = α
lims ′n = β.
7/25/2019 Series2014 I
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{x n}
n ,
a
= x
, an+
= x n+
− x n. ∑ an
{x n
}
∑ +∞n=
cos (nx )np
p >
x ∈ R.
− .
