Alcuni sviluppi di McLaurin notevoli

x2 x3 xn =1+x+ + + Â·Â·Â· + + o (xn ) 2! 3! n!

x

e

sinh x

=x+

cosh x

=1+

4

2n

x x x + + Â·Â·Â· + 2! 4! (2n)!

  1 2 = x â&#x2C6;&#x2019; x3 + x5 + o x6 3 15

tanh x ln (1 + x)

sin x

=1â&#x2C6;&#x2019;

tan x arcsin x arccos x arctan x Î±

  + o x2n+1

n [ xk

k=0 n [ k=0 n [ k=0

  x3 x5 x2n+1 + + Â· Â· Â· + (â&#x2C6;&#x2019;1)n + o x2n+2 3! 5! (2n + 1)!

2n   x2 x4 n x + + Â· Â· Â· + (â&#x2C6;&#x2019;1) + o x2n+1 2! 4! (2n)!

  1 2 = x + x3 + x5 + o x6 3 15    â&#x2C6;&#x2019;1/2  x2n+1   1 3 3 5   = x + x + x + Â·Â·Â· +  + o x2n+2  6 40 2n + 1 n =

= =

x2 x3 x4 xn =xâ&#x2C6;&#x2019; + â&#x2C6;&#x2019; + Â· Â· Â· + (â&#x2C6;&#x2019;1)nâ&#x2C6;&#x2019;1 + o (xn ) 2 3 4 n =xâ&#x2C6;&#x2019;

cos x

=

  x3 x5 x2n+1 + + Â·Â·Â· + + o x2n+2 3! 5! (2n + 1)! 2

(1 + x)

(si sottintende ovunque che i resti sono trascurabili per x â&#x2020;&#x2019; 0)

Ï&#x20AC; â&#x2C6;&#x2019; arcsin x 2

  x3 x5 x2n+1 =xâ&#x2C6;&#x2019; + + Â· Â· Â· + (â&#x2C6;&#x2019;1)n + o x2n+2 3 5 2n + 1       Î± 2 Î± 3 Î± n = 1 + Î±x + x + x + Â·Â·Â· + x + o (xn ) 2 3 n

=

=

=

k=0 n [

  x2k+1 + o x2n+2 (2k + 1)!   x2k + o x2n+2 (2k)!

(â&#x2C6;&#x2019;1)

kâ&#x2C6;&#x2019;1

(â&#x2C6;&#x2019;1)

k

(â&#x2C6;&#x2019;1)k

xk + o (xn ) k

  x2k+1 + o x2n+2 (2k + 1)!   x2k + o x2n+1 (2k)!

 n  [  â&#x2C6;&#x2019;1/2  x2k+1     + o x2n+2 =   k 2k + 1 k=0

=

=

= 1 â&#x2C6;&#x2019; x + x2 â&#x2C6;&#x2019; x3 + x4 + Â· Â· Â· + (â&#x2C6;&#x2019;1)n xn + o (xn )

=

1 1â&#x2C6;&#x2019;x

= 1 + x + x2 + x3 + x4 + Â· Â· Â· + xn + o (xn )

=

â&#x2C6;&#x161; 1+x

1 1 = 1 + x â&#x2C6;&#x2019; x2 + 2 8

1 â&#x2C6;&#x161; 1+x

1 3 = 1 â&#x2C6;&#x2019; x + x2 â&#x2C6;&#x2019; 2 8

â&#x2C6;&#x161; 3 1+x

1 1 = 1 + x â&#x2C6;&#x2019; x2 + 3 9

1 â&#x2C6;&#x161; 3 1+x

1 2 = 1 â&#x2C6;&#x2019; x + x2 â&#x2C6;&#x2019; 3 9

n fattori

k=1 n [

+ o (xn )

k=0

1 1+x

  1 3 1/2 n x + o (xn ) x + Â·Â·Â· + n 16   5 3 â&#x2C6;&#x2019;1/2 n x + o (xn ) x + Â·Â·Â· + n 16   5 3 1/3 n x + o (xn ) x + Â·Â·Â· + n 81   7 3 â&#x2C6;&#x2019;1/3 n x + o (xn ) x + Â·Â·Â· + n 81

n [

k!

= = = =

n [

(â&#x2C6;&#x2019;1)

k=0 n  [ k=0 n [ k=0 n [

k

  x2k+1 + o x2n+2 2k + 1

 Î± k x + o (xn ) k k

(â&#x2C6;&#x2019;1) xk + o (xn ) xk + o (xn )

k=0 n  [

 1/2 k x + o (xn ) k k=0  n  [ â&#x2C6;&#x2019;1/2 k x + o (xn ) k k=0  n  [ 1/3 k x + o (xn ) k k=0  n  [ â&#x2C6;&#x2019;1/3 k x + o (xn ) k k=0

Â&#x20AC; ~     } Î± Î± Î± (Î± â&#x2C6;&#x2019; 1) Â· Â· Â· (Î± â&#x2C6;&#x2019; n + 1) Si ricordi che â&#x2C6;&#x20AC;Î± â&#x2C6;&#x2C6; R si pone =1e = se n â&#x2030;¥ 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...