Series de Taylor-Maclaurin
f(x) =
∞
∑
n=0
f(n)(0)
n!xn=f(0) + f0(0)x+f00(0)
2! x2+f000(0)
3! x3+···+f(n)(0)
n!xn+···
Función Desarrollo de Taylor-Maclaurin Válido para
1
1−x
∞
∑
n=0
xn=1+x+x2+···+xn+··· x∈(−1,1)
1
1+x
∞
∑
n=0
(−1)nxn=1−x+x2+···+ (−1)nxn+··· x∈(−1,1)
1
√1+x
∞
∑
n=0−1/2
nxn=1−1
2x+1·3
2·4x2−1·3·5
2·4·6x3+···+ (−1)n(2n−1)!!
(2n)!! xn+··· x∈(−1,1)
(1+x)α
∞
∑
n=0α
nxn=1+αx+α(α−1)
2! x2+···+α(α−1). . . (α−n+1)
n!xn+··· x∈(−1,1)
log(1+x)
∞
∑
n=1
(−1)n+1
nxn=x−1
2x2+1
3x3−1
4x4+···+(−1)n+1
nxn+··· x∈(−1,1]
ex
∞
∑
n=0
1
n!xn=1+x+1
2x2+1
3! x3+1
4! x4+···+1
n!xn+··· x∈R
senx
∞
∑
n=0
(−1)n
(2n+1)!x2n+1=x−1
3! x3+1
5! x5−1
7! x7+···+(−1)n
(2n+1)!x2n+1+··· x∈R
cosx
∞
∑
n=0
(−1)n
(2n)!x2n=1−1
2! x2+1
4! x4−1
6! x6+···+(−1)n
(2n)!x2n+··· x∈R
tgx
∞
∑
n=0
tg(2n+1)(0)
(2n+1)!x2n+1=x+1
3x3+2
15 x5+17
315 x7+62
2835 x9+···+tg(2n+1)(0)
(2n+1)!x2n+1+··· x∈(−
π
2,π
2)
arcsenx
∞
∑
n=0
(2n−1)!!
(2n)!!(2n+1)x2n+1=x+1
6x3+3
40 x5+5
112 x7+···+(2n−1)!!
(2n)!!(2n+1)x2n+1+··· x∈[−1,1]
arctgx
∞
∑
n=0
(−1)n
2n+1x2n+1=x−1
3x3+1
5x5−1
7x7+···+(−1)n
2n+1x2n+1+··· x∈[−1,1]
Brook TAYLOR Colin M ACL AUR IN
1685 – 1731 1698 – 1746