Special Power Series

ex = 1 + x +x2

2!+

x3

3!+ ... +

xr

r!+ ... (all x)

sin x = x − x3

3!+

x5

5!− x7

7!+ ... +

(−1)rx2r+1

(2r + 1)!+ ... (all x)

cos x = 1 − x2

2!+

x4

4!− x6

6!+ ... +

(−1)rx2r

(2r)!+ ... (all x)

tan x = x +x3

3+

2x5

15+

17x7

315+ ... (|x| < π

2)

sin−1 x = x +1

2

x3

3+

1.3

2.4

x5

5+

1.3.5

2.4.6

x7

7+

... +1.3.5....(2n − 1)

2.4.6....(2n)

x2n+1

2n + 1+ ... (|x| < 1)

tan−1 x = x − x3

3+

x5

5− x7

7+ ... + (−1)n x2n+1

2n + 1+ ... (|x| < 1)

�n(1 + x) = x − x2

2+

x3

3− x4

4+ ... + (−1)n+1 xn

n+ ... (−1 < x ≤ 1)

sinh x = x +x3

3!+

x5

5!+

x7

7!+ ... +

x2n+1

(2n + 1)!+ ... (all x)

cosh x = 1 +x2

2!+

x4

4!+

x6

6!+ ... +

x2n

(2n)!+ ... (all x)

tanh x = x − x3

3+

2x5

15− 17x7

315+ ... (|x| < π

2)

sinh−1 x = x − 1

2

x3

3+

1.3

2.4

x5

5− 1.3.5

2.4.6

x7

7+

... + (−1)n 1.3.5...(2n − 1)

2.4.6...2n

x2n+1

2n + 1+ ... (|x| < 1)

tanh−1 x = x +x3

3+

x5

5+

x7

7+ ...

x2n+1

2n + 1+ ... (|x| < 1)

2

Stat 401: Introduction to Probability Handout-05, September 27, 2006

Source: http://www.maths.manchester.ac.uk/~cds/internal/tables/tables.html