Alcuni sviluppi di McLaurin notevoli (si sottintende ovunque che i resti sono trascurabili per x→ 0)

ex = 1 + x+x2

2!+x3

3!+ · · ·+ x

n

n!+ o (xn) =

n

k=0

xk

k!+ o (xn)

sinhx = x+x3

3!+x5

5!+ · · ·+ x2n+1

(2n+ 1)!+ o x2n+2 =

n

k=0

x2k+1

(2k + 1)!+ o x2n+2

coshx = 1 +x2

2!+x4

4!+ · · ·+ x2n

(2n)!+ o x2n+1 =

n

k=0

x2k

(2k)!+ o x2n+2

tanhx = x− 13x3 +

2

15x5 + o x6

ln (1 + x) = x− x2

2+x3

3− x

4

4+ · · ·+ (−1)n−1 x

n

n+ o (xn) =

n

k=1

(−1)k−1 xk

k+ o (xn)

sinx = x− x3

3!+x5

5!+ · · ·+ (−1)n x2n+1

(2n+ 1)!+ o x2n+2 =

n

k=0

(−1)k x2k+1

(2k + 1)!+ o x2n+2

cosx = 1− x2

2!+x4

4!+ · · ·+ (−1)n x2n

(2n)!+ o x2n+1 =

n

k=0

(−1)k x2k

(2k)!+ o x2n+1

tanx = x+1

3x3 +

2

15x5 + o x6

arcsinx = x+1

6x3 +

3

40x5 + · · ·+ −1/2

n

x2n+1

2n+ 1+ o x2n+2 =

n

k=0

−1/2k

x2k+1

2k + 1+ o x2n+2

arccosx =π

2− arcsinx

arctanx = x− x3

3+x5

5+ · · ·+ (−1)n x

2n+1

2n+ 1+ o x2n+2 =

n

k=0

(−1)k x2k+1

2k + 1+ o x2n+2

(1 + x)α = 1 + αx+α

2x2 +

α

3x3 + · · ·+ α

nxn + o (xn) =

n

k=0

α

kxk + o (xn)

1

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

n

k=0

(−1)k xk + o (xn)

1

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

n

k=0

xk + o (xn)

√1 + x = 1 +

1

2x− 1

8x2 +

1

16x3 + · · ·+ 1/2

nxn + o (xn) =

n

k=0

1/2

kxk + o (xn)

1√1 + x

= 1− 12x+

3

8x2 − 5

16x3 + · · ·+ −1/2

nxn + o (xn) =

n

k=0

−1/2k

xk + o (xn)

3√1 + x = 1 +

1

3x− 1

9x2 +

5

81x3 + · · ·+ 1/3

nxn + o (xn) =

n

k=0

1/3

kxk + o (xn)

13√1 + x

= 1− 13x+

2

9x2 − 7

81x3 + · · ·+ −1/3

nxn + o (xn) =

n

k=0

−1/3k

xk + o (xn)

Si ricordi che ∀α ∈ R si pone α

0= 1 e

α

n=

n fattori

α (α− 1) · · · (α− n+ 1)n! se n ≥ 1.