1. AN ´ALISIS MATEM ´ATICO - PRIMER CURSO - UNIVERSIDAD DE ZARAGOZA
F´ORMULA DE TAYLOR (DESARROLLOS LIMITADOS)
f(x) = f(a) + f (a)(x − a) +
f (a)
2!
(x − a)2
+ . . . +
f(n
(a)
n!
(x − a)n
+ Rn(x);
Rn(x) =
f(n+1
(t)
(n + 1)!
(x − a)n+1
= o((x − a)n
), x → a
EJEMPLOS:
1)
1
1 − x
= 1 + x + x2
+ x3
+ . . . + xn
+
1
(1 − t)n+2
xn+1
.
2) ex
= 1 + x +
1
2!
x2
+
1
3!
x3
+ . . . +
1
n!
xn
+
et
(n + 1)!
xn+1
.
3) log(1 + x) = x −
1
2
x2
+
1
3
x3
−
1
4
x4
+ . . . +
(−1)n−1
n
xn
+
(−1)n
(n + 1)(1 + t)n+1
xn+1
.
4) (1 + x)α
= 1 + αx +
α
2
x2
+ . . . +
α
n
xn
+
α
n + 1
(1 + t)α−n−1
xn+1
.
5) sen x = x −
1
3!
x3
+
1
5!
x5
−
1
7!
x7
+ . . . +
(−1)n
(2n + 1)!
x2n+1
+
(−1)n+1
cos t
(2n + 3)!
x2n+3
.
6) cos x = 1 −
1
2!
x2
+
1
4!
x4
−
1
6!
x6
+ . . . +
(−1)n
(2n)!
x2n
+
(−1)n+1
cos t
(2n + 2)!
x2n+2
.
7) tg x = x +
1
3
x3
+
2
15
x5
+
17
315
x7
+ o(x8
), cuando x → 0.
8) sec x = 1 +
1
2
x2
+
5
24
x4
+
61
720
x6
+ o(x7
), cuando x → 0.
9) arc sen x = x +
1
6
x3
+
3
40
x5
+
5
112
x7
+ . . . +
1 · 3 · 5 · . . . · (2n − 1)
2 · 4 · 6 · . . . · (2n)
·
x2n+1
2n + 1
+ o(x2n+2
),
cuando x → 0.
10) arc tg x = x −
1
3
x3
+
1
5
x5
−
1
7
x7
+ . . . +
(−1)n
2n + 1
x2n+1
+ o(x2n+2
), cuando x → 0.
11) senh x = x +
1
3!
x3
+
1
5!
x5
+
1
7!
x7
+ . . . +
1
(2n + 1)!
x2n+1
+
cosh t
(2n + 3)!
x2n+3
.
12) cosh x = 1 +
1
2!
x2
+
1
4!
x4
+
1
6!
x6
+ . . . +
1
(2n)!
x2n
+
cosh t
(2n + 2)!
x2n+2
.
13) tgh x = x −
1
3
x3
+
2
15
x5
−
17
315
x7
+ o(x8
), cuando x → 0.
14) arg senh x = x −
1
6
x3
+
3
40
x5
+ . . . + (−1)n 1 · 3 · 5 · . . . · (2n − 1)
2 · 4 · 6 · . . . · (2n)
·
x2n+1
2n + 1
+ o(x2n+2
),
cuando x → 0.
15) arg tgh x = x +
1
3
x3
+
1
5
x5
+
1
7
x7
+ . . . +
1
2n + 1
x2n+1
+ o(x2n+2
), cuando x → 0.