Page 1

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

Sviluppi_McLaurin  

x 2n+1 2n+1+o x 2n+2 = 1 1−x =1+x+x 2 +x 3 +x 4 +···+x n +o(x n ) = k x k +o(x n ) 3 x 3 +···+ α √1+x =1+1 2x− √1+x =1+1 3x− 1 8x 2 + 1...

Read more
Read more
Similar to
Popular now
Just for you