Taylor Series for Functions of One Variable TAYLOR SERIES
22 Taylor Series for Functions of One Variable TAYLOR SERIES
n − f 1 ( n ′′ −− ( )( ax − a ) f ( )( ax a )
22.1. fx () = fa () +′ fax ( )( − a )
where R n , the remainder after n terms, is given by either of the following forms:
f () n ( )( ξ x − a ) n
22.2. Lagrange’s form: R n =
f () n ( )( x
ξ − ξ )( x − a )
22.3. Cauchy’s form: R
( n − 1 )!
The value x, which may be different in the two forms, lies between a and x. The result holds if f(x) has continuous derivatives of order n at least.
If lim n →∞ R n = 0 the infinite series obtained is called the Taylor series for f(x) about x ⫽ a. If a ⫽ 0, the series , is often called a Maclaurin series. These series, often called power series, generally converge for all values of x
in some interval called the interval of convergence and diverge for all x outside this interval. Some series contain the Bernoulli numbers B n and the Euler numbers E n defined in Chapter 23, pages 142⫺143.
Binomial Series
n − 1 nn ( − 1 ) n − 22 nn ( − 1 )( n − 2 22.4. ) ( a + x ) = a + na x +
Special cases are
22.5. ( a + x ) 2 = a 2 + 2 ax + x 2
22.6. ( a + x ) 3 = a 3 + 3 ax 2 + 3 ax 2 + x 3
22.7. ( a + x ) 4 = a 4 + 4 ax 3 + 6 ax 22 + 4 ax 3 + x 4
22.8. ( 1 + x ) − 1 =−+ 1 x x 2 − x 3 + x 4 − ⋅⋅⋅
⫺ 1<x<1
22.9. ( 1 + x ) − 2 =− 12 x + 3 x 2 − 4 x 3 + 5 x 4 − ⋅⋅⋅
⫺ 1<x<1
22.10. ( 1 + x ) − 3 =− 13 x + 6 x 2 − 10 x 3 + 15 x 4 − ⋅⋅⋅
⫺ 1<x<1
TAYLOR SERIES
1 13 i
135 i i
22.11. ( 1 + x ) − 12 / =− 1 x +
2 24 246 i i + ⋅⋅⋅
x 3 ⫺ 1<x⬉1
1 1 2 13 i
22.12. ( 1 + x ) =+ 1 x −
36 ⫺ i − 3 1<x⬉1
Series for Exponential and Logarithmic Functions
22.15. e =++ 1 x
22.16. a x
22.17. ln ( 1 + x ) =− x
22.19. ln x 2 ⎪ ⎛ − ⎞
22.20. ln x
⎝⎜ x ⎠⎟ 2 ⎝⎜ x ⎠⎟ 3 ⎝⎜ x ⎠⎟
Series for Trigonometric Functions
22.21. sin x =− x + − + ⋯ − ∞< < ∞ x
22.22. cos x =− 1 + − + ⋯ − ∞< < ∞ x
x 3 2 x 5 17 x 7 2 2 n ( 2 2 n 1 ) Bx 2 n − 11
22.23. tan x =+ x
22.24. cot x
22.25. sec x =+ 1 ⋯
720 + + ( )! 2 n +
1 x 7 x 3 31 x 5 22 ( 2 n − 1 1 ) B xx 2 n − 1
=++ n + + ⋯ + + ⋯ 0 < || x x 6 360 , 15 120 ( )! 2 n < π − 1 1 x 3 13 i x 5 135 i i x 22.27. 7 sin x =+ x
22.26. csc x
23 + 245 i + 2467 i i + ⋯ || x <1
22.28. cos − 1 x π
1 x 3 13 i x 5
= 2 − sin − 1 x = 2 − x +
|| x <1
TAYLOR SERIES
|| x < 1
22.29. tan − 1 x =
22.31. sec − 1 x cos ( / ) − 1 1 x π
22.32. csc − 1 x sin ( / ) − 1
Series for Hyperbolic Functions
22.33. sinh x =+ x + + + ⋯
− ∞< < ∞ x
22.34. cosh x =+ 1 + + + ⋯
− ∞< < ∞ x
x 3 2 x 5 17 x 7 () 1 n − 12 2 n ( 2 2 n 1 )) Bx 2 n − 1 π
22.35. tanh x x
22.36. coth x
22.37. sech x 1 ⋯ −
22.38. csch x =−+ x 6 360 −
− 1 ⎪⎪ 23 i 245 i i 2467 i ii
|| | x < 1
22.39. sinh x = ⎨ ⎛
⎡ if cosh x 0 ,⭌ x 1 ⎤
22.40. cosh − 1
i i i x 6 ⎬ ⎩⎪ − ⎝⎜ 24 66 6 1 ⎠⎟ ⎣⎢ − if cosh x < 0 , x ⭌ ⎭⎪ 1 ⎦⎥ − 1 x 3 x 5 x 22.41. 7 tanh x =+ x ⋯ | x |<1
Miscellaneous Series
sin x
22.43. e =++ 1 x
2 − 8 − 15 + ⋯ −∞< < ∞ x
22.44. e cos x e ⎛ 1 = ⎞ −
x 2 x 4 31 x 6
−∞< < ∞ x
TAYLOR SERIES
22.45. e tan x =++ 1 x
2 + 2 + 8 + ⋯ || x < 2
x 3 x 5 x 6 2 n / x 2 sin( n π /) 4 x n
22.46. e sin x =+ x x 2 + −
x x 3 x 4 2 n / 2 cos( n /) 4 x n
22.47. e cos x =+− 1 x
x 2 x 4 x 6 2 2 n − 1 Bx 2 n n
22.48. ln | sin | x = n ln | | x − ⋯
− 1 Bx 22.49. n ln | cos | n =−
22.50. ln | tan | ln | | x = x + +
22.51. + x =−+ x ( 1 1 ) x 2 +++ ( 1 1 1 3
|| x <1
Reversion of Power Series Suppose
22.52. yCxCx = + 2 + Cx 3 + Cx 4 + Cx 5 1 6 2 3 4 5 + Cx 6 + ⋯
then
22.53. xCyCy = 1 + 2 2 + Cy 3 3 + Cy 4 4 + Cy 5 5 + Cy 6 6 + ⋯
where
22.54. cC 11 = 1
22.55. cC 1 3 2 =− c 2
22.56. cC 5 1 2 3 = 2 c 2 − cc 13
22.57. cC 1 7 4 = 5 ccc 5 c 123 2 − 3 2 − cc 1 4
22.58. cC 9 = 6 ccc 2 + 3 cc 2 1 2 1 3 cc 3 4 1 2 5 24 − 1 5 + 14 c 2 − 21 ccc 12 33
22.59. cC 11 = 7 ccc 3 + 84 ccc 3 + 7 ccc 3 − 28 ccc 2 2 4 2 1 2 6 1 25 12 3 1 34 1 2 33 − cc 1 6 − 28 ccc 1 2 4 − 42 c 5 2
Taylor Series for Functions of Two Variables