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ANALISIS MATEMATICO - PRIMER CURSO - UNIVERSIDAD DE ZARAGOZA
FORMULA 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 +12!
x2 +13!
x3 + . . . +1n!
xn +et
(n + 1)!xn+1.
3) log(1 + x) = x− 12x2 +
13x3 − 1
4x4 + . . . +
(−1)n−1
nxn +
(−1)n
(n + 1)(1 + t)n+1xn+1.
4) (1 + x)α = 1 + αx +(
α
2
)x2 + . . . +
(α
n
)xn +
(α
n + 1
)(1 + t)α−n−1xn+1.
5) senx = x− 13!
x3 +15!
x5 − 17!
x7 + . . . +(−1)n
(2n + 1)!x2n+1 +
(−1)n+1 cos t
(2n + 3)!x2n+3.
6) cos x = 1− 12!
x2 +14!
x4 − 16!
x6 + . . . +(−1)n
(2n)!x2n +
(−1)n+1 cos t
(2n + 2)!x2n+2.
7) tg x = x +13x3 +
215
x5 +17315
x7 + o(x8), cuando x → 0.
8) sec x = 1 +12x2 +
524
x4 +61720
x6 + o(x7), cuando x → 0.
9) arc senx = x+16x3 +
340
x5 +5
112x7 + . . .+
1 · 3 · 5 · . . . · (2n− 1)2 · 4 · 6 · . . . · (2n)
· x2n+1
2n + 1+ o(x2n+2),
cuando x → 0.
10) arc tg x = x− 13x3 +
15x5 − 1
7x7 + . . . +
(−1)n
2n + 1x2n+1 + o(x2n+2), cuando x → 0.
11) senhx = x +13!
x3 +15!
x5 +17!
x7 + . . . +1
(2n + 1)!x2n+1 +
cosh t
(2n + 3)!x2n+3.
12) coshx = 1 +12!
x2 +14!
x4 +16!
x6 + . . . +1
(2n)!x2n +
cosh t
(2n + 2)!x2n+2.
13) tghx = x− 13x3 +
215
x5 − 17315
x7 + o(x8), cuando x → 0.
14) arg senhx = x− 16x3 +
340
x5 + . . . + (−1)n 1 · 3 · 5 · . . . · (2n− 1)2 · 4 · 6 · . . . · (2n)
· x2n+1
2n + 1+ o(x2n+2),
cuando x → 0.
15) arg tghx = x +13x3 +
15x5 +
17x7 + . . . +
12n + 1
x2n+1 + o(x2n+2), cuando x → 0.