Sviluppi di McLaurin
ex
=
2
3
4
n
1 + x + x + x + x + · · · + x + o(xn )
2!
3!
4!
n!
(x → 0)
sinh x
=
3
5
2n+1
x + x + x + ··· + x
+ o(x2n+2 )
3!
5!
(2n + 1)!
(x → 0)
cosh x
=
2
4
2n
1 + x + x + ··· + x
+ o(x2n+1 )
2!
4!
(2n)!
(x → 0)
tanh x
=
3
2 x5 + o(x6 )
x − x3 + 15
(x → 0)
sett tanh x
=
3
5
x2n+1 + o(x2n+2 )
x + x3 + x5 + · · · + 2n
+1
(x → 0)
ln(1 + x)
=
n
2
3
x − x2 + x3 + · · · + (−1)(n−1) xn + o(xn )
(x → 0)
sin x
=
2n+1
3
5
x − x + x + · · · + (−1)n x
+ o(x2n+2 )
3!
5!
(2n + 1)!
(x → 0)
cos x
=
2n
2
4
1 − x + x + · · · + (−1)n x
+ o(x2n+1 )
2!
4!
(2n)!
(x → 0)
tan x
=
3
2 x5 + o(x6 )
x + x3 + 15
(x → 0)
arcsin x
=
3
3 x5 + o(x6 )
x + x6 + 40
(x → 0)
arctan x
=
3
2n+1
x − x3 + 15 x5 + · · · + (−1)n x
+ o(x2n+2 )
(2n + 1)
(x → 0)
µ
α
α
2
¶
µ
2
¶
µ
3
¶
xn + o(xn )
=
1 + αx +
1
(1 + x)
=
1 − x + x2 − x3 + · · · + (−1)n xn + o(xn )
(x → 0)
1
(1 − x)
=
1 + x + x2 + x3 + · · · + xn + o(xn )
(x → 0)
=
1 x3 + · · · +
1 + 12 x − 18 x2 + 16
(1 + x)
p 1
(1 + x)
p
3
=
(1 + x)
=
1
p
3
(1 + x)
=
x + ··· +
α
n
(1 + x)
p
x +
α
3
µ
1/2
n
¶
xn + o(xn )
¶
−1/2
xn + o(xn )
n
µ
¶
5 x3 + · · · + 1/3 xn + o(xn )
1 + 13 x − 19 x2 + 81
n
µ
¶
−1/3
2
7
1
2
3
1 − 3 x + 9 x − 81 x + · · · +
xn + o(xn )
n
5 x3 + · · · +
1 − 12 x + 38 x2 − 16
1
(x → 0)
(x → 0)
µ
(x → 0)
(x → 0)
(x → 0)