Machinery's Handbook, 31st Edition
Series
135
Series Representation of a Function Some hand calculations, as well as computer programs of certain types of mathematical problems, may be facilitated by the use of an appropriate series. For example, in some gear problems, the angle corresponding to a given or calculated involute function (see TRIGONOMETRY ) is found by using a series together with an iterative procedure such as the Newton-Raphson method described in the previous section. The following are those series most commonly used for such purposes. In the series for trigonometric func- tions, the angles x are in radians (1 radian = 180/ p degrees, or about 57.3 degrees). The expression exp( - x 2 ) means that the base e of the natural logarithm system is raised to the - x 2 power, where e = 2.7182818. A sum of terms in a sequence of terms is called a series . In its simplest notation, the in- finite sum of the sequence a 0 , a 1 , a 2 , . . . is the series a a a a n n = ∞ = + + + 0 0 1 2 ... ∑ . For example, for the sequence of x , x 2 , x 3 , . . . , the series is given as x x x x x n n = ∞ = + + = + + 1 1 2 1 2 ... ... ∑ . In calculus, it is helpful to represent certain functions by a special series called the Taylor series. In a manner similar to the Newton-Raphson method, the terms of a Taylor series include the derivatives of the function being approximated. In the table below, common functions and their corresponding infinite series are shown. For any x in the domain (shown to the right of each series), the value of the function can be found. Common Series sin cos ! ! ! ! ! ! x x x x x x x x x x = − + − + = − + − 3 5 7 2 4 6 3 5 7 2 4 6 1 for all real + = + + − + < = − for all real for x x x x x x x x 3 5 7 3 2 15 17 315 2 π 1 x x x x 3 5 5 3 15
tan sin cos
x x x
x x + + + + x 3 5 6 40 336 3
for for for
≤ ≤
1 1 1
−
1
5
2 π
= − π
x = − + + + + 15 (
−
1
x
x
)
sin
2
6
40
336
−
1
= − + − + x x x x x 3 5 7
tan
≤
3 5 7
π 4
= − + − 1 + = + + + = + + + + + = 1 7 1 3 1 5 1 1 1 2 3 1 2 3 x x x ! ! ! ! ! 1 1 1 1 ! + + ln ! 1 x a x a ( ln ) ! 2 2
e e
x
x
for all real
+ + x a ( ln ) ! 3 3
x
a
x
for all real
+ = − + x x x ) 2
x x x 3 4 5 3 4 5 − +
x 1 1
− < ≤
for for
−
ln(
1
2
1
x x 2 3 4 − +
x x
x
1 = − +
−
1
≤
x
+
1
1
x x 2 3 4
x = + + 1
x
x
for
1
+ + +
≤
x 1
−
1
2
3
4
x x x x 1 2 3 4 5 = − + − + −
x
≤
for
1
2
(
)
x
+
1
1
+ + + + + 2 3 4 5 2 3 4 x x x x
x
for
1
=
1
≤
2
x − ( ) 1
1 3 ∙
2
3
1 3 5 ∙ ∙
2 x
x
x
1
≤ 1 x for
1 = − +
−
−
...
∙
2 4 6 ∙ ∙
x
+
1
2 4
1 ( ) x n na x n n a x − na n n n 1 2 2 3 3 ( ) ( ) − − − − −
2 1 3 )( !
(
+ a x a ) = n
n
+
x
+
+
+
...
for all real
1
!
2
!
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