Esercizi svolti di analisi reale e complessa - Dipartimento di ...

α > 0
• [0, 1)
• [0, a] 0 < a < 1
A ≡ { 1
n 2 | n ≥ 1}
[−1, 1] \ A
([−1, a] ∪
[0, 1]) ∩ A c −1 < a < 0
(−e −1 , e −1 )
K ⊂ (−e −1 , e −1 )
(0, +∞)
[a, +∞) a > 0
∀x ∈ (−1, 1),
n≥1 n3xn = x3 +4x 2 +x
(1−x) 4
x = 1
2
n≥1 n3
2n = 26
x |x| < 1
1
1−x
k≥0 xk = 1
1
1−x j 1−x
dj
dx j ( 1
1−x
) =
j!
(1−x) j+1
j!
|x| < 1 (1−x) j+1 = (k+j)!
k≥0 k! xk 1
∀ |x| < 1 :
k≥0
k + n − 1
n − 1
(1−x) j+1 =
x m
(1+x) n m 1 = x (1−x) n
m k + n − 1
= x k≥0
n − 1
xm+k =
n − 1 − m + k
k≥m
x
j
k
k + j
j
xk =
e x = +∞
k=0 xk
k!
∀ x ∈ R e x = +∞
k=0 xk
k!
e x
|e x −
N
k=0
xk |x|N+1
| ≤
k! (N + 1)! eξ N→∞
−→ 0 ∃ξ
limn→∞
k≥0
√
n!
2πn( n ) n = 1
e
(xe) n
√ 2πn+x
x k