Sviluppi di Taylor e Mac Laurin Analisi Matematica I Natali Mattia
Sviluppi di Taylor e Mac Laurin
Sviluppo di Taylor:
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Tn = f (x0) + f '(x0)⋅ (x − x0) + f ''(x0)⋅ (x − x0)2
2! + ...+ f(n )(x0)
n! (x − x0)n.
Sviluppi di Mac Laurin se x0 = 0.
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ex = 1+ x +x2 2! + x3
3! + x4
4! + ...+xn
n! + o(xn)
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ln(1+ x) = x − x2 2 + x3
3 − x4
4 + ...+ (−1)n +1xn
n + o(xn)
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(1+ x)a = 1+ ax +a(a −1)
2! x2+a(a −1)(a − 2)
3! x3+ ...+ a n
⎛
⎝ ⎜ ⎞
⎠ ⎟ xn+ o(xn)
con
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a =1 2
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1+ x = 1+ x 2 − x2
8 + x3
16 + o(x3)
con
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a = −1
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(1+ x)−1= 1 − x + x2− x3+ ...+ (−1)nxn + o(xn)
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sin x = x − x3 3!+ x5
5! + ...+ (−1)n x2n +1
(2n +1)!+ o(x2n +2)
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cos x = 1 − x2 2! +x4
4! + ...+ (−1)n x2n
(2n)!+ o(x2n +1)
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tan x = x + x3 3 + 2
15x5+ o(x6)
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arcsin x = x + x3 6 + 3
40x5+ o(x6)
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arctan x = x − x3 3 + x5
5 + ...+ (−1)n x2n +1
2n +1+ o(x2n +2)
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sinh x = x + x3 3! +x5
5! + ...+ x2n +1
(2n +1)!+ o(x2n +2)
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cosh x = 1+ x2 2! + x4
4!+ ...+ x2n
(2n)!+ o(x2n +1)
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tanh x = x − x3 3 + 2
15x5+ o(x6)
