hp40g+.book Page 25 Friday, December 9, 2005 1:03 AM

Typing:

 

 

 

 

 

ERIES

⎛

COS(2

2

π

⎞

⎝

⋅ X)

, X = --, 4

 

 

 

6

⎠

gives:

1

 

 

2

 

8 3

 

3

8

4

 

⎛h5⎞

h = X –

π--〉

〈4--

–

3h + 2h

 

+

------3---

h

 

– 3--h

 

+ 0

⎝----4 ⎠

 

 

 

 

 

 

 

 

 

 

 

 

 

6

•Example — Expansion in the vicinity of x=+∞ or x=–∞

Example 1

Give a 5th-order expansion of arctan(x) in the vicinity of

x=+∞, taking as infinitely small

h =

1

 

-- .

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

Typing:

 

 

 

 

 

 

 

 

 

 

 

 

SERIES(ATAN(X),X =+∞,5)

 

 

 

gives:

 

 

 

 

 

 

 

 

 

 

 

 

 

⎛π

h3

h5

+ 0

⎛

π

⋅ h6⎞ ⎞

 

 

 

1

 

 

 

 

 

--

– h + ---- –

----

⎝

------------

h =

 

⎝ 2

3

5

 

 

2

⎠ ⎠

--

 

Example 2

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

----------

 

Give a 2nd-order expansion of (2x – 1)ex – 1

in the

vicinity of x=+∞, taking as infinitely small h =

1

-- .

 

 

 

 

 

 

 

 

 

 

 

 

 

x

ERIES((2X – 1 ) ⋅ EXP

⎛ 1

⎞

 

 

 

 

------------

, X = + ∞, 3)

 

 

 

 

 

 

 

⎝X – 1⎠

 

 

 

 

gives:

 

 

 

 

 

 

 

 

 

 

 

 

 

12

+ 6h + 12h2 + 17h3

+

0(2

⋅ h

3

)

 

1

 

 

 

------------------------------------------------------

 

 

 

 

6 ⋅ h

 

 

 

 

 

 

 

 

 

 

h = --

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

•Unidirectional expansion

To perform an expansion in the vicinity of x = a where x > a, use a positive real (such as 4.0) for the order.

To perform an expansion in the vicinity of x = a where x < a, use a negative real (such as –4.0) for the order.

Computer Algebra System (CAS)

14-25