Fórmulas Matemáticas

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 
 
 
 
 
 
𝑠 = 𝑎1 + 𝑎2 +⋯+ 𝑎𝑛 +⋯
lim
𝑛→∞
(𝑎1 +⋯+ 𝑎𝑛)
(𝑎𝑛)
(𝑠𝑛)
𝑠1 = 𝑎1, 𝑠2 = 𝑎1 + 𝑎2, ⋯ , 𝑠𝑛 = 𝑎1 + 𝑎2 +⋯+ 𝑎𝑛, ⋯
𝑠𝑛 ∑𝑎𝑛
𝑎𝑛 𝒏 −
𝑠 = lim
𝑛→∞
𝑠𝑛 ∑𝑎𝑛
𝑠 = ∑𝑎𝑛 =∑ 𝑎𝑛
∞
𝑛=1 = 𝑎1 + 𝑎2 +⋯+ 𝑎𝑛 +⋯
∑𝑎𝑛
 
1 + 𝑎 + 𝑎2 +⋯+ 𝑎𝑛 +⋯
|𝑎| < 1 1 (1 − 𝑎)⁄ 0 < 𝑎 < 1
𝑠𝑛 = 1 + 𝑎 + 𝑎
2 +⋯+ 𝑎𝑛 𝑠𝑛 =
(1 − 𝑎𝑛+1)
(1 − 𝑎)⁄ 𝑠𝑛−1 < 𝑠𝑛
𝑠𝑛 <
1
(1 − 𝑎)⁄ 𝑛 ∈ ℕ
lim
𝑛→∞
(1 (1 − 𝑎)⁄ − 𝑠𝑛) = lim𝑛→∞
𝑎𝑛
(1 − 𝑎)⁄ = 0,
lim
𝑛→∞
𝑠𝑛 = lim ( 1 + 𝑎 + 𝑎
2 +⋯+ 𝑎𝑛) = 1 (1 − 𝑎)⁄
−1 < 𝑎 < 1 lim
𝑛→∞
𝑠𝑛 =0
|𝑎| < 1 lim |𝑎|𝑛 = 0 lim 𝑎𝑛 = 0
1 + 1 + 1 2!⁄ +⋯+
1
𝑛!⁄ + ⋯
𝑒
𝑠𝑛 = 1 + 1 +
1
2!⁄ + ⋯+
1
𝑛!⁄
2 ≤ 𝑠𝑛 ≤ 1 + 1 +
1
2
+
1
22
+⋯+
1
2𝑛
< 3,
2 ≤ 𝑠𝑛
𝑢𝑛 = 1 +
1
2⁄ +
1
22⁄ ⋯+
1
2𝑛⁄
 
𝑢𝑛 =
1 − (1 2⁄ )
𝑛+1
1 − 1 2⁄
= 2 − 1 2𝑛⁄ ,
𝑠𝑛 = 1 + 𝑢𝑛 = 3 −
1
2𝑛⁄ 𝑠𝑛 < 3
𝑒 =
lim
𝑛→∞
𝑠𝑛
𝑒 = 2.7182…
1 − 1 + 1 − 1 +⋯ (−1)𝑛+1
𝑠𝑛 𝑛 1 𝑛
lim 𝑠𝑛
∑1 𝑛(𝑛 + 1)⁄ 𝑎𝑛 = 1/𝑛(𝑛 + 1) = 1/𝑛 −
 1/(𝑛 + 1) 𝑛 − 
𝑠𝑛 = (1 −
1
2
) + (
1
2
−
1
3
) +⋯+ (
1
𝑛
−
1
𝑛 + 1
) = 1 −
1
𝑛 + 1
.
 
lim (𝑠𝑛) = 1 ∑
1
𝑛(𝑛 + 1)⁄ = 1
∑𝑎𝑛 𝑎𝑛 ≥ 0 𝑛 ∈ ℕ ∑𝑎𝑛
∑𝑎𝑛
𝑘 𝑎1 +⋯+ 𝑎𝑛 ≤ 𝑘
𝑛 ∈ ℕ ∑𝑎𝑛 ≤ +∞
∑𝑎𝑛 𝑎𝑛 ≥ 0
𝑎𝑛 ≥ 0 𝑛 ∈ ℕ (𝑎′𝑛) (𝑎𝑛)
∑ 𝑎𝑛 ≤ +∞ ∑𝑎′𝑛 ≤ +∞
∑1 𝑛⁄
∑
1
𝑛
= 𝑠
∑
1
2𝑛
= 𝑡 ∑
1
2𝑛−1
= 𝑢 𝑠𝑛 =
𝑡𝑛 + 𝑢𝑛
𝑛 → ∞ 𝑠 = 𝑡 + 𝑢 𝑡 =
∑
1
2𝑛
=
1
2
∑
1
𝑛
=
𝑠
2
𝑢 = 𝑡 =
𝑠
2
𝑢 − 𝑡 = lim
𝑛→∞
(𝑢𝑛 − 𝑡𝑛)
 = lim
𝑛→∞
[(1 −
1
2
) + (
1
3
−
1
4
) +⋯+ (
1
2𝑛 − 1
−
1
2𝑛
)]
 = lim
𝑛→∞
(
1
1 ∙ 2
+
1
3 ∙ 4
+ ⋯+
1
(2𝑛 − 1) ∙ 2𝑛
) > 0
𝑢 > 𝑡
 
∑𝑎𝑛 ∑𝑏𝑛
0 𝑐 > 0 𝑛0 ∈ ℕ 𝑎𝑛 ≤ 𝑐𝑏𝑛
𝑛 > 𝑛0 ∑𝑏𝑛 ∑𝑎𝑛
∑𝑎𝑛 ∑𝑏𝑛
𝑎𝑛 ≤ 𝑐𝑏𝑛
𝑛 ∈ ℕ
𝑐
𝑛 ∈ ℕ 𝑐′ = 𝑐 +
𝑎1
𝑏1
⁄ +
⋯+
𝑎𝑛0−1
𝑏𝑛0−1
⁄
𝑎𝑛 ≤ 𝑐𝑏𝑛 𝑛 ∈ ℕ
𝑠𝑛 𝑡𝑛 ∑𝑎𝑛 ∑𝑏𝑛
𝑠𝑛 ≤ 𝑐𝑡𝑛 𝑛 >
𝑛0 𝑐 > 0 (𝑡𝑛) (𝑠𝑛) (𝑠𝑛)
(𝑡𝑛) 𝑡𝑛 ≥ 𝑠𝑛/𝑐 ■
 
𝑟 > 1 ∑1 𝑛𝑟⁄
𝑐 ∑ (2 2𝑟⁄ )
𝑛∞
𝑛=0
𝑠𝑛 ∑
1
𝑛𝑟
𝑐 𝑛
𝑚 ≤ 2𝑛 − 1
𝑠𝑚 ≤ 1 + (
1
2𝑟
+
1
3𝑟
) + (
1
4𝑟
+
1
5𝑟
+
1
6𝑟
+
1
7𝑟
) + 
+⋯+ (
1
(2𝑛−1)𝑟
+⋯+
1
(2𝑛 − 1)𝑟
) ,
𝑠𝑚 < 1 +
2
2𝑟
+
4
4𝑟
+⋯+
2𝑛−1
(2𝑛−1)𝑟
= ∑(
2
2𝑟
)
𝑖
< 𝑐
𝑛−1
𝑖=0
∑
1
𝑛𝑟
𝑟 > 1 1 𝑛𝑟⁄ >
1
𝑛⁄
∑𝑎𝑛 𝑠𝑛 = 𝑎1 +⋯+ 𝑎𝑛 s = lim
𝑛→∞
𝑠𝑛
(𝑡𝑛) 𝑡1 = 0 𝑡𝑛 = 𝑠𝑛−1 lim 𝑡𝑛 = 𝑠 𝑠𝑛 −
𝑡𝑛 = 𝑎𝑛 lim 𝑎𝑛 = lim (𝑠𝑛 − 𝑡𝑛) = lim (𝑠𝑛) − lim (𝑡𝑛) = 𝑠 − 𝑠 = 0
■
 
∑𝑎𝑛
∑ |𝑎𝑛|
−1 < 𝑎 < 1 ∑ 𝑎𝑛∞𝑛=0
∑ |𝑎|𝑛∞𝑛=0 |𝑎
𝑛| = |𝑎|𝑛
−1 < 𝑎 < 1
∑ (−1)
𝑛+1
𝑛⁄ = 1 −
1
2
+
1
3
−
1
4
+⋯∞𝑛=0
(𝑎𝑛)
∑(−1)𝑛+1𝑎𝑛
𝑠𝑛 = 𝑎1 − 𝑎2 +⋯+ (−1)
𝑛+1𝑎𝑛 𝑠2𝑛 = 𝑠2𝑛−2 + 𝑎2𝑛−1 − 𝑎2𝑛 𝑠2𝑛+1 =
𝑠2𝑛−1 − 𝑎2𝑛 + 𝑎2𝑛+1
𝑎2𝑛−1 − 𝑎2𝑛 ≥ 0
−𝑎2𝑛 + 𝑎2𝑛+1 ≤ 0 𝑠2𝑛 =
𝑠2𝑛−1 − 𝑎2𝑛 𝑠2𝑛−1 − 𝑠2𝑛 = 𝑎2𝑛 ≥ 0
 
𝑠2 ≤ 𝑠4 ≤ ⋯ ≤ 𝑠2𝑛−1 ≤ ⋯ ≤ 𝑠3 ≤ 𝑠1
𝑠2𝑛 = lim 𝑠2𝑛−1 lim 𝑎𝑛 = 0 (𝑠𝑛)
■
∑(−1)𝑛+1 log (1 +
1
𝑛
)
𝑛 − ∑ log (1 +
1
𝑛
) = ∑ log (
𝑛+1
𝑛
)
 𝑠𝑛 = log 2 + log (
3
2
) + log (
4
3
) +⋯+ log (
𝑛 + 1
𝑛
)
 
= log 2 + log 3 − log 2 + log 4 − log 3 +⋯+ log(𝑛 + 1) − log(𝑛)
= log(𝑛 + 1). 
lim 𝑠𝑛 = +∞
∑𝑎𝑛 ∑ |𝑎𝑛| = +∞
 
0
∑ |𝑎𝑛| 𝑛
𝑝𝑛 𝑞𝑛
𝑝𝑛 = {
𝑎𝑛, 𝑠𝑖 𝑎𝑛 ≥ 0
 0, 𝑠𝑖 𝑎𝑛 < 0
𝑞𝑛 = {
−𝑎𝑛, 𝑠𝑖 𝑎𝑛 ≤ 0
 0, 𝑠𝑖 𝑎𝑛 > 0
𝑝𝑛 𝑞𝑛
𝑎𝑛 𝑛 ∈ ℕ 𝑝𝑛 𝑞𝑛
(𝑝𝑛 , 𝑞𝑛 ≥ 0 (𝑝𝑛 , 𝑞𝑛 ≤ |𝑎𝑛|) 𝑝𝑛 + 𝑞𝑛 = |𝑎𝑛|
∑ 𝑝𝑛 ∑𝑞𝑛
∑𝑎𝑛 = ∑(𝑝𝑛 − 𝑞𝑛) = ∑𝑝𝑛 −∑𝑞𝑛 ■
 
∑𝑎𝑛
∑𝑝𝑛 = +∞ ∑𝑞𝑛 = +∞
∑𝑎𝑛 = ∑𝑝𝑛 − ∑𝑞𝑛 = 𝑎 −∞ =
−∞ ∑ |𝑎𝑛| = ∑𝑝𝑛 + ∑𝑞𝑛 < +∞
∑𝑏𝑛 𝑏𝑛 ≠ 0
𝑛 ∈ ℕ (𝑎𝑛 𝑏𝑛⁄ )
∑𝑎𝑛
𝑐 > 0 |
𝑎𝑛
𝑏𝑛
⁄ | ≤ 𝑐 𝑛 ∈ ℕ
|𝑎𝑛| ≤ 𝑐|𝑏𝑛|
∑ 𝑎𝑛
■
𝑎𝑛 ≠ 0 𝑛 ∈ ℕ
𝑐 | 𝑎𝑛+1 𝑎𝑛⁄ | ≤ 𝑐 < 1 𝑛
lím |𝑎𝑛+1 𝑎𝑛⁄ | < 1 ∑𝑎𝑛
 
𝑛
|𝑎𝑛+1 𝑎𝑛⁄ | ≤ 𝑐 =
𝑐𝑛+1
𝑐𝑛⁄ ,
|𝑎𝑛+1|
𝑐𝑛+1
⁄ ≤
|𝑎𝑛|
𝑐𝑛⁄ .
|𝑎𝑛|
𝑐𝑛⁄ ≥ 0 𝑛 ∈ ℕ
∑𝑐𝑛 ∑𝑎𝑛
lím |𝑎𝑛+1| |𝑎𝑛⁄ | = 𝐿 < 1 𝑐
𝐿 < 𝑐 < 1 |𝑎𝑛+1| |𝑎𝑛⁄ | < 𝑐 𝑛
𝑎 = lim𝑥𝑛 𝑏 < 𝑎
𝑛 𝑏 < 𝑥𝑛 𝑎 < 𝑏
𝑥𝑛 < 𝑏 𝑛
■
lím |𝑎𝑛+1 𝑎𝑛⁄ | = 𝐿
𝐿 > 1 |𝑎𝑛+1 𝑎𝑛⁄ | > 1
|𝑎𝑛+1| > |𝑎𝑛| 𝑛 𝑎𝑛
 
𝐿 = 1
∑1
𝑛2⁄
∑1 𝑛⁄
𝑐 √|𝑎𝑛|
𝑛 < 1
𝑛 ∈ ℕ lím √|𝑎𝑛|
𝑛 < 1
∑𝑎𝑛
√|𝑎𝑛|
𝑛 ≤ 𝑐 < 1 |𝑎𝑛| ≤ 𝑐
𝑛 𝑛
∑ 𝑐𝑛
∑𝑎𝑛
lim √|𝑎𝑛|
𝑛
= 𝐿 < 1 𝑐 𝐿 < 𝑐 < 1
√|𝑎𝑛|
𝑛 < 𝑐 𝑛
■
lim √|𝑎𝑛|
𝑛
= 𝐿 𝐿 > 1
lim √|𝑎𝑛|
𝑛
> 1 𝑛
|𝑎𝑛| > 1 ∑𝑎𝑛
𝐿 = 1 ∑1 𝑛2⁄
∑1 𝑛⁄
 
𝑎𝑛 = (
log 𝑛
𝑛⁄ )
𝑛
√𝑎𝑛
𝑛 =
log
𝑛⁄ ∑𝑎𝑛
(𝑎𝑛)
lím |𝑎𝑛+1| |𝑎𝑛|⁄ = 𝐿 lím √|𝑎𝑛|
𝑛
= 𝐿
𝑎𝑛 > 0 𝑛 ∈ ℕ
𝐿 ≠ 0 𝜀 > 0 𝐾 𝑀 𝐿 − 𝜀 < 𝐾 < 𝐿 < 𝑀 < 𝐿 + 𝜀
𝑝 𝑛 ≥ 𝑝 ⟹ 𝐾 <
𝑎𝑛+1
𝑎𝑛⁄ < 𝑀
𝑛 − 𝑝 𝐾 <
𝑎𝑝+𝑖
𝑎𝑝+𝑖−1⁄ < 𝑀 𝑖 = 1,⋯ , 𝑛 − 𝑝
𝐾𝑛 <
𝑎𝑛
𝑎𝑝⁄ < 𝑀
𝑛−𝑝 𝑛 > 𝑝 𝛼 =
𝑎𝑝
𝐾𝑝⁄ 𝛽 =
𝑎𝑝
𝑀𝑝⁄ 𝐾
𝑛𝛼 < 𝑎𝑛 < 𝑀
𝑛𝛽 𝐾√𝛼
𝑛
< √𝛼
𝑛
< 𝑀√𝛽
𝑛
𝑛 > 𝑝 𝐿 − 𝜀 < 𝐾 𝑀 < 𝐿 + 𝜀 lim √𝛼
𝑛
= 1
lim √𝛽
𝑛 = 1 𝑟0 > 𝑝 𝑛 > 𝑛0 ⟹ 𝐿 − 𝜀 < 𝐾√𝛼
𝑛
𝑀√𝛽
𝑛 < 𝐿 + 𝜀 𝑛 > 𝑛0 ⟹ 𝐿− 𝜀 < √𝛼
𝑛
< 𝐿 + 𝜀
𝐿 = 0 𝑀
𝐿 ≠ 0
■
 
lim𝑛
√𝑛!
𝑛⁄ =
𝑒 𝑎𝑛 =
𝑛𝑛
𝑛!⁄
𝑛
√𝑛!
𝑛⁄
𝑎𝑛+1
𝑎𝑛
=
(𝑛 + 1)𝑛+1
(𝑛 + 1)!
𝑛!
𝑛𝑛
=
(𝑛 + 1)(𝑛 + 1)𝑛
(𝑛 + 1)𝑛!
𝑛!
𝑛 ∙ 𝑛
= (
𝑛 + 1
𝑛
)
𝑛
lim(
𝑎𝑛+1
𝑎𝑛⁄ ) = 𝑒 lim √𝑎𝑛
𝑛 = 𝑒
∑𝑎𝑛
𝜑:ℕ → ℕ 𝑏𝑛 = 𝑎𝜑(𝑛) ∑𝑏𝑛
∑𝑎𝑛 ∑𝑏𝑛 = ∑𝑎𝑛
𝜑
∑𝑎𝑛
𝑠 = 1 −
1
2
+
1
3
−
1
4
+⋯
 
𝑠
2
=
1
2
−
1
4
+
1
6
−
1
8
+⋯.
𝑠 = 1 −
1
2
+
1
3
−
1
4
+
1
5
−
1
6
+
1
7
−
1
8
+⋯.
𝑠
2
= 0 −
1
2
+ 0 −
1
4
+ 0 −
1
6
+ 0 −
1
8
+⋯.
3𝑠
2
= 1 +
1
3
−
1
2
+
1
5
+
1
7
−
1
4
+
1
9
+
1
11
−
1
6
⋯.
3𝑠
2
𝑠
∑𝑎𝑛
𝜑:ℕ → ℕ 𝑏𝑛 = 𝑎𝜑(𝑛) ∑𝑏𝑛 = ∑𝑎𝑛.
𝑎𝑛 ≥ 0 𝑠𝑛 = 𝑎1 +⋯+ 𝑎𝑛 𝑡𝑛 = 𝑏1 +
⋯+ 𝑏𝑛 𝑛 ∈ ℕ 𝜑(1),⋯ , 𝜑(𝑛)
{1,2,⋯ ,𝑚} 𝑚 𝜑(𝑖)
𝑡𝑛 =∑𝑏𝑗 ≤
𝑛
𝑗=1
∑𝑎𝑗
𝑚
𝑗=1
= 𝑠𝑚.
 
𝑛 ∈ ℕ 𝑚 ∈ ℕ 𝑠𝑚 < 𝑡𝑛
lim 𝑡𝑛 < lim 𝑠𝑛 ∑𝑏𝑛 = ∑𝑎𝑛
∑𝑎𝑛 = ∑𝑝𝑛 − ∑𝑞𝑛 𝑝𝑛 𝑞𝑛
𝑎𝑛 (𝑏𝑛)
𝑎𝑛 (𝑢𝑛) 𝑝𝑛 (𝑣𝑛) 𝑞𝑛
𝑢𝑛 𝑣𝑛 𝑏𝑛
∑𝑢𝑛 = ∑𝑝𝑛 ∑𝑣𝑛 = ∑𝑞𝑛
∑𝑎𝑛 =∑𝑢𝑛
−∑𝑣𝑛 =∑𝑏𝑛
■
∑𝑎𝑛 𝑐
∑𝑎𝑛
 
𝑎𝑛1 𝑐
∑𝑎𝑛 +∞
𝑎𝑛2 𝑐
−∞
∑𝑎𝑛
𝑐 𝑛1
𝑎𝑛𝑘
lim
𝑘→∞
𝑎𝑛𝑘 = 0 ∑𝑎𝑛
𝑐
 
𝜀 − 𝛿
 
http://www.biografiasyvidas.com/biografia/w/weierstrass.htm
http://www.biografiasyvidas.com/biografia/w/weierstrass.htm
http://www.biografiasyvidas.com/biografia/w/weierstrass.htm
 
𝑓: 𝑋 ⟶ ℝ 𝑋 ⊆ ℝ
𝑎 ∈ 𝑋
𝜀 > 0 𝛿 > 0 𝑎 ∈ 𝑋 |𝑥 − 𝑎| < 𝛿
𝑓(𝑥) − 𝑓(𝑎)| < 𝜀
𝑓 𝑎
∀ 𝜀 > 0 ∃ 𝛿 > 0 ∋ 𝑎 ∈ 𝑋 y |𝑥 − 𝑎| < 𝛿 ⟹ 𝑓(𝑥) − 𝑓(𝑎)| < 𝜀.
𝑓 𝐴 ⊆ ℝ 𝑓 𝑎,
𝑎 ∈ 𝐴
 
𝑓: 𝑋 ⟶ ℝ
𝑋 𝜀 > 0
𝛿 > 0 𝑥, 𝑦 ∈ 𝑋 |𝑦 − 𝑥| < 𝛿 𝑓(𝑦) −
𝑓(𝑥)| < 𝜀
𝑓
𝑋
∀ 𝜀 > 0 ∃ 𝛿 > 0 ∋ 𝑥, 𝑦 ∈ 𝑋 y |𝑦 − 𝑥| < 𝛿 ⟹ 𝑓(𝑦) − 𝑓(𝑥)|
< 𝜀.
 𝑓: 𝑋 ⟶ ℝ
𝑋
𝛿 𝜀 𝑎
𝜀
 [a, b] ⊆ ℝ f: [a, b] → ℝ,
[a, b] f [a, b]
 
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ, 𝑓
𝐼𝑚𝑓
𝑓: [−1,2] → ℝ, 
𝑓(𝑥) =
{
 
 
 
 
−2 si − 1 ≤ 𝑥 < 0
1
2⁄ si 0 ≤ 𝑥 < 1
 0 si 1 ≤ 𝑥 <
1
2
 1 si 
1
2
≤ 𝑥 < 2
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ, [𝑎, 𝑏] 𝜀 > 0,
𝑓𝜀 ∶ [𝑎, 𝑏] → ℝ |𝑓(𝑥) − 𝑓𝜀(𝑥)| < 𝜀
𝑥𝜖[𝑎, 𝑏].
𝑓: [𝑎, 𝑏] → ℝ,
[𝑎, 𝑏] 𝜀 > 0, 𝛿 > 0 𝑥, 𝑦 ∈ [𝑎, 𝑏]
 
|𝑥 − 𝑦| < 𝛿, |𝑓(𝑥) − 𝑓(𝑦)| < 𝜀 𝑚𝜖ℕ ℎ =
𝑏−𝑎
𝑚
< 𝛿. [𝑎, 𝑏] 𝑚 ℎ,
𝐼1 = [𝑎, 𝑎 + ℎ], 𝐼2 = [𝑎 + ℎ, 𝑎 + 2ℎ], … , 𝐼𝑘 = [𝑎 + (𝑘 − 1)ℎ, 𝑎 + 𝑘ℎ], … , 𝐼𝑚 = [𝑎 + (𝑚 −
1)ℎ, 𝑎 + 𝑚ℎ = 𝑏]
𝐼𝑘 ℎ < 𝛿, 
𝑓 𝐼𝑘 𝜀
𝑓𝜀: [𝑎, 𝑏] → 𝑅 𝑓𝜀(𝑥) = 𝑓(𝑎 + 𝑘ℎ) 𝑥 ∈ 𝐼𝑘 𝑘 = 1,2, …𝑚
𝐼𝑘
𝑥 ∈ 𝐼𝑘
|𝑓(𝑥) − 𝑓𝜀(𝑥)| = |𝑓(𝑥) − 𝑓(𝑎 + 𝑘ℎ)| < 𝜀
|𝑓(𝑥) − 𝑓𝜀(𝑥)| < 𝜀 𝑥𝜖[𝑎, 𝑏].
■
 
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ,
𝑓 [𝒂, 𝒃], [𝑎, 𝑏]
𝐼1, 𝐼2, … 𝐼𝑚 𝑓 𝐼𝑘
 
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ, [𝑎, 𝑏] 𝜀 > 0,
𝑓𝜀 ∶ [𝑎, 𝑏] → ℝ |𝑓(𝑥) − 𝑓𝜀(𝑥)| <
𝜀 𝑥𝜖[𝑎, 𝑏].
𝑓 [𝑎, 𝑏]
 𝜀 > 0, 𝛿 > 0 𝑥, 𝑦 ∈ [𝑎, 𝑏] |𝑥 − 𝑦| < 𝛿, |𝑓(𝑥) − 𝑓(𝑦)| <
𝜀 𝑚 ∈ ℕ ℎ =
𝑏−𝑎
𝑚
< 𝛿, 
[𝑎, 𝑏] 𝑚 ℎ: 𝐼1 = [𝑎, 𝑎 + ℎ], … 𝐼𝑘 = [𝑎 +
(𝑘 − 1)ℎ, 𝑎 + 𝑘ℎ], 𝑘 = 2,… ,𝑚 𝐼𝑘 𝑓𝜀
(𝑎 + (𝑘 − 1)ℎ, 𝑓(𝑎 + (𝑘 − 1)ℎ)) (𝑎 + 𝑘ℎ, 𝑓(𝑎 + 𝑘ℎ)) . 
𝑓𝜀 [𝑎, 𝑏],
𝑥 ∈ 𝐼𝑘 𝑓(𝑥) 𝜀 
𝑓(𝑎 + (𝑘 − 1)ℎ) 𝑓(𝑎 + 𝑘ℎ), |𝑓(𝑥) − 𝑓𝜀(𝑥)| < 𝜀
𝑥𝜖[𝑎, 𝑏].
■
 
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ,
[𝑎, 𝑏] 𝜀 > 0, 𝑝𝜀: [𝑎, 𝑏] → ℝ
|𝑓(𝑥) − 𝑝𝜀(𝑥)| < 𝜀 𝑥𝜖[𝑎, 𝑏].
𝐴 ⊆ ℝ ∀ 𝑛 ∈ ℕ
𝑓𝑛: 𝐴 ⟶ ℝ {𝑓𝑛} 𝐴 ℝ.
 
{𝑓𝑛(𝑥)} 𝑓(𝑥) 𝐴 ⊆ ℝ, 𝑓
{𝑓𝑛(𝑥)} 𝐴, ∀𝒂 ∈ 𝐴
lim
𝑛→∞
𝑓𝑛(𝑎) = 𝑓(𝑎)
lim
𝑛→∞
𝑓𝑛(𝑥) =
𝑝
 𝑓(𝑥) 𝑓𝑛 ⟶
𝑝
𝑓 
{𝑓𝑛} 𝑓
𝐴
{𝑓𝑛(𝑥)} 𝑓(𝑥) 𝐴 ⊆ ℝ, 𝑓
{𝑓𝑛(𝑥)} 𝐴,
∀𝜀 > 0, ∃𝑁 ∈ ℕ 𝑛 > 𝑁 ⇒ |𝑓𝑛(𝑥) − 𝑓(𝑥)| < 𝜀 ∀𝑥
∈ 𝐴
lim
𝑛→∞
𝑓𝑛(𝑥) =
𝑢
 𝑓(𝑥) 𝑓𝑛 ⟶
𝑢
𝑓
{𝑓𝑛} 
𝑓 𝐴
𝑎 ∈ 𝐴
𝑓,
𝑓(𝑎)
{𝑓𝑛(𝑎)} 𝑓(𝑎)
∀𝜀 > 0, ∃𝑁 ∈ ℕ 𝑛 > 𝑁 ⇒ |𝑓𝑛(𝑎) − 𝑓(𝑎)| < 𝜀
 
𝑁 𝜀 𝑎.
𝑁 𝜀
 
𝑓(𝑥) = |𝑥|
𝑏 ∈ (0,1), 𝑎 ∈ ℤ 𝑎𝑏 >
1 +
3
2
𝜋, 
 
𝑓(𝑥) = ∑ 𝑏𝑛 cos(𝑎𝑛𝜋𝑥)∞𝑛=0 ∀𝑥 ∈ ℝ.
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ,
[𝑎, 𝑏] 𝜀 > 0, 𝑝𝜀: [𝑎, 𝑏] → ℝ
|𝑓(𝑥) − 𝑝𝜀(𝑥)| < 𝜀 𝑥 ∈ [𝑎, 𝑏].
 
[0,1]
𝑓: [0,1] → ℝ,
𝐵𝑛(𝑥) =∑𝑓(
𝑘
𝑛
) (
𝑛
𝑘
) 𝑥𝑘(1 − 𝑥)𝑛−𝑘
𝑛
𝑘=0
 
𝐵𝑛(𝑥) 𝒏 − 𝒇;
𝑛,
 𝑓 𝑛 + 1 [0,1] 
0,
1
𝑛
,
2
𝑛
,
3
𝑛
, … ,
𝑛−1
𝑛
, 1
 (𝑛
𝑘
) =
𝑛!
𝑘!(𝑛−𝑘)!
𝑓: [0,1] → ℝ, 𝜀 >
0, 𝑁 ∈ ℕ 𝑛 ≥ 𝑁, |𝑓(𝑥) − 𝐵𝑛(𝑥)| < 𝜀 𝑥 ∈ [0,1].
 
 
 [𝑎, 𝑏] ⊆ ℝ 𝑓: [𝑎, 𝑏] → ℝ, [𝑎, 𝑏].
𝑔: [0,1] → [𝑎, 𝑏] 𝑔(𝑡) = 𝑎 + 𝑡(𝑏 − 𝑎) 𝑡 ∈ [0,1].
 
𝑔 𝑔(0) = 𝑎 𝑔(1) = 𝑏. 𝑓 
𝑓 ∘ 𝑔 𝜀 > 0,
𝑁 𝑛 ≥ 𝑁, 𝑛 −
𝐵𝑛(𝑓 ∘ 𝑔) 
|(𝑓 ∘ 𝑔)(𝑥) − 𝐵𝑛(𝑓 ∘ 𝑔)(𝑥)| < 𝜀 𝑥 ∈ [0,1] __________________________________(1)
𝑔 𝑔−1: [𝑎, 𝑏] →
[0,1]
𝑔−1(𝑡) =
𝑡 − 𝑎
𝑏 − 𝑎
𝑡 ∈ [𝑎, 𝑏].
 
𝑡 ∈ [𝑎, 𝑏]
|𝑓(𝑡) − 𝐵𝑁(𝑓 ∘ 𝑔)( 𝑔
−1(𝑡))| < 𝜀.
|𝑓(𝑡) − 𝐵𝑁(𝑓 ∘ 𝑔) ( 
𝑡 − 𝑎
𝑏 − 𝑎
)| < 𝜀 𝑡 ∈ [𝑎, 𝑏].
𝐵𝑁(𝑓 ∘ 𝑔) ( 
𝑡−𝑎
𝑏−𝑎
) [𝑎, 𝑏].
𝑝𝜀: [𝑎, 𝑏] → ℝ 𝑝𝜀(𝑥) = 𝐵𝑁(𝑓 ∘ 𝑔) ( 
𝑥−𝑎
𝑏−𝑎
) |𝑓(𝑥) −
𝑝𝜀(𝑥)| < 𝜀 𝑥 ∈ [𝑎, 𝑏].
■
𝑞𝑛(𝑥) = 𝐵𝑛(𝑓 ∘ 𝑔) ( 
𝑥−𝑎
𝑏−𝑎
) 𝑓 
[𝑎, 𝑏].
 
■
𝑞𝑛(𝑥) 𝐵𝑛(𝑓 ∘ 𝑔) ( 
𝑥 − 𝑎
𝑏 − 𝑎
)
∑(𝑓 ∘ 𝑔) (
𝑘
𝑛
) (
𝑛
𝑘
) ( 
𝑥 − 𝑎
𝑏 − 𝑎
)
𝑘
(1
𝑛
𝑘=0
− ( 
𝑥 − 𝑎
𝑏 − 𝑎
))
𝑛−𝑘
∑(𝑓 ∘ 𝑔) (
𝑘
𝑛
) (
𝑛
𝑘
) ( 
𝑥 − 𝑎
𝑏 − 𝑎
)
𝑘
( 
𝑏 − 𝑥
𝑏 − 𝑎
)
𝑛−𝑘
𝑛
𝑘=0
∑𝑓(𝑎 +
𝑘
𝑛
(𝑏
𝑛
𝑘=0
− 𝑎)) (
𝑛
𝑘
) ( 
𝑥 − 𝑎
𝑏 − 𝑎
)
𝑘
( 
𝑏 − 𝑥
𝑏 − 𝑎
)
𝑛−𝑘
𝑓 [𝑎, 𝑏].
 
ℝ 
ℝ. 
𝐶([𝑎, 𝑏], ℝ)
𝑋 𝐴 ⊆ 𝑋
𝐶(𝐴,ℝ)
𝐴 𝑑∞
𝐶(𝐴,ℝ)
𝑓, 𝑔 ∈ 𝐶(𝐴, ℝ) α ∈ ℝ ⇒ 𝑓 + 𝑔, 𝑓𝑔, α𝑓 ∈ 𝐶(𝐴,ℝ). 
ℝ
𝐶(𝐴,ℝ) 
𝐶(𝐴,ℝ) 
 
𝐶(𝐴,ℝ) 𝐴 
𝑋 𝐴 ⊆ 𝑋
𝐶(𝐴, ℝ) 𝐴.
𝐵 𝐶(𝐴,ℝ)
∀𝑥, 𝑦 ∈ 𝐴 ⇒ ∃𝑓 ∈ 𝐵 𝑓(𝑥) ≠ 𝑓(𝑦), 𝐵 
𝐶(𝐴,ℝ).
 
file:///C:/Users/malo__000/Downloads/DESCARGABLE/MAMT2U1Evidencia.docx
file:///C:/Users/malo__000/Downloads/DESCARGABLE/MAMT2U1Evidencia.docx
 
 
ℝ
 
http://www.miscelaneamatematica.org/Misc25/grabinsky.pdf
 
http://matematicas.unex.es/~montalvo/Analisis_Varias_Variables/apuntes/indice.pdf
http://matematicas.unex.es/~montalvo/Analisis_Varias_Variables/apuntes/indice.pdf
http://www.miscelaneamatematica.org/Misc25/murillo.pdf