Alcuni sviluppi di McLaurin notevoli
x2 x3 xn =1+x+ + + ··· + + o (xn ) 2! 3! n!
x
e
sinh x
=x+
cosh x
=1+
4
2n
x x x + + ··· + 2! 4! (2n)!
1 2 = x − x3 + x5 + o x6 3 15
tanh x ln (1 + x)
sin x
=1−
tan x arcsin x arccos x arctan x α
+ o x2n+1
n [ xk
k=0 n [ k=0 n [ k=0
x3 x5 x2n+1 + + · · · + (−1)n + o x2n+2 3! 5! (2n + 1)!
2n x2 x4 n x + + · · · + (−1) + o x2n+1 2! 4! (2n)!
1 2 = x + x3 + x5 + o x6 3 15 −1/2 x2n+1 1 3 3 5 = x + x + x + ··· + + o x2n+2 6 40 2n + 1 n =
= =
x2 x3 x4 xn =x− + − + · · · + (−1)n−1 + o (xn ) 2 3 4 n =x−
cos x
=
x3 x5 x2n+1 + + ··· + + o x2n+2 3! 5! (2n + 1)! 2
(1 + x)
(si sottintende ovunque che i resti sono trascurabili per x → 0)
π − arcsin x 2
x3 x5 x2n+1 =x− + + · · · + (−1)n + o x2n+2 3 5 2n + 1 α 2 α 3 α n = 1 + αx + x + x + ··· + x + o (xn ) 2 3 n
=
=
=
k=0 n [
x2k+1 + o x2n+2 (2k + 1)! x2k + o x2n+2 (2k)!
(−1)
k−1
(−1)
k
(−1)k
xk + o (xn ) k
x2k+1 + o x2n+2 (2k + 1)! x2k + o x2n+1 (2k)!
n [ −1/2 x2k+1 + o x2n+2 = k 2k + 1 k=0
=
=
= 1 − x + x2 − x3 + x4 + · · · + (−1)n xn + o (xn )
=
1 1−x
= 1 + x + x2 + x3 + x4 + · · · + xn + o (xn )
=
√ 1+x
1 1 = 1 + x − x2 + 2 8
1 √ 1+x
1 3 = 1 − x + x2 − 2 8
√ 3 1+x
1 1 = 1 + x − x2 + 3 9
1 √ 3 1+x
1 2 = 1 − x + x2 − 3 9
n fattori
k=1 n [
+ o (xn )
k=0
1 1+x
1 3 1/2 n x + o (xn ) x + ··· + n 16 5 3 −1/2 n x + o (xn ) x + ··· + n 16 5 3 1/3 n x + o (xn ) x + ··· + n 81 7 3 −1/3 n x + o (xn ) x + ··· + n 81
n [
k!
= = = =
n [
(−1)
k=0 n [ k=0 n [ k=0 n [
k
x2k+1 + o x2n+2 2k + 1
α k x + o (xn ) k k
(−1) xk + o (xn ) xk + o (xn )
k=0 n [
1/2 k x + o (xn ) k k=0 n [ −1/2 k x + o (xn ) k k=0 n [ 1/3 k x + o (xn ) k k=0 n [ −1/3 k x + o (xn ) k k=0
€ ~ } α α α (α − 1) · · · (α − n + 1) Si ricordi che ∀α ∈ R si pone =1e = se n ≥ 1. n! 0 n