+ ⋅⋅⋅⎫⎬⎪ ⎭⎪
ln
x
=
⎛ ⎝⎜
x
− 1⎞ x ⎠⎟
+
1 2
⎛x ⎝⎜
− 1⎞ 2 x ⎠⎟
+
1 3
⎛ ⎝⎜
x
− 1⎞ 3 x ⎠⎟
+
⋅⋅⋅
Ϫ∞ < x < ∞ Ϫ∞ < x < ∞ Ϫ1 < x Ϲ 1 Ϫ1 < x < 1 x>0 xм1
2
Series for Trigonometric Functions
Some
series
contain
the
Bernoulli
numbers
B n
and
the
Euler
numbers
E n
defined
in
Chapter
23,
pages
142Ϫ143.
Binomial Series
22.4.
(a
+
x)n
=
an
+
nan−1x
+
n(n − 1) 2!
an−2 x2
+
n(n
−
−
22n−1(22n − 1)Bn x 2n n(2n)!
+
ln | tan x
|
=
ln |
x
|+
x2 3
+
7x4 90
+
62 x 6 2835
+
+
22n (22n−1 − 1)Bn x 2n n(2n)!
+
ln(1 + x) 1+ x
=
x
−
(1 +
1 2
)
x
2
+
(1 +
1 2
+
1 3
)x
3
−
|
x
ecos x
=
e ⎛⎝⎜1 −
x2 2
+
x4 6
−
31x 6 720
+
⎞ ⎠⎟
− ∞< x < ∞
− ∞< x < ∞
|
x
|
<
π 2
0< |x| <π
|
x
|
<
π 2
0< |x| <π
|x| <1 ⎡ + if x м 1 ⎤ ⎣⎢− if x Ϲ − 1⎦⎥ ⎡+ if cosh−1 x > 0, x м 1⎤ ⎣⎢− if cosh−1 x < 0, x м 1⎦⎥ |x|<1
3 4
i i
5x7 6i7
⎨ ⎪± ⎩⎪
⎛ ⎝⎜ln
|
2x
|
+
1 2 i 2x2
−
1i3 2 i 4 i 4x4
+ +
2
1i3i5 i 4 i 6 i 6x6
−
⎞ ⎠⎟
22.40. 22.41. 22.42.
cosh−1 x
=
±
⎧⎨⎪ln(2x) ⎩⎪
−
⎛ ⎝⎜
2
i
1 2
x
2
+
2
1i 3 i 4 i 4x4
tanh x
=
x
−
x3 3
+
2x5 15
−
17x 7 315
+
(−1)n−1 22n (22n − 1)Bn x 2n−1 (2n)!
+
22.36.
coth
x
=
1 x
+
x 3
−
x3 45
+
2x5 945
+
(−1)n−1 22n Bn x 2n−1 (2n)!
+
22.37.
sech
x
=1−
x2 2
+
5x4 24
(1 + x)1/2
=1 +
1 2
x
−
1 2i4
x2
+
1i3 2i4i6
x3
− ⋅⋅⋅
(1 +
x )−1/ 3
=1
−
1 3
x+
1i 4 3i6
x2
−
1i 4 i 7 3i6i9
x3
+ ⋅⋅⋅
(1 +
x )1/ 3
=1
+
1 3x
−
2 3i6
x2
+
2i5 3i6i9
x3
− ⋅⋅⋅
Ϫ1 < x Ϲ 1 Ϫ1 < x Ϲ 1 Ϫ1 < x Ϲ 1 Ϫ1 < x Ϲ 1
|x|>1
−∞< x < ∞ −∞< x < ∞
22.45. 22.46. 22.47. 22.48. 22.49. 22.50. 22.51.
etan x
=1+
x
+
x2 2
+
x3 2
+
3x 4 8
+
ex sin x = x + x2 + x3 − x5 − x6 + + 2n/2 sin (nπ /4)xn +
− 1)(n − 3!
2)
an−3 x 3
+ ⋅⋅⋅
=
an
+
⎛n⎞ ⎝⎜1⎠⎟
a n−1 x
+
⎛n⎞ ⎝⎜2⎠⎟
an−2 x2
+
⎛n⎞ ⎝⎜ 3⎠⎟
an−3 x 3
+
⋅⋅⋅
Special cases are
22.5. (a + x)2 = a2 + 2ax + x2
22.6. (a + x)3 = a3 + 3a2 x + 3ax2 + x3
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.
22.2.
Lagrange’s form:
Rn
=
f (n) (ξ)(x − a)n n!
22.3.
Cauchy’s form:
Rn
=
f (n) (ξ)(x − ξ)n−1(x − a) (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
Ϫ1 < x < 1
22.10. (1 + x)−3 = 1 − 3x + 6x2 − 10x3 + 15x4 − ⋅ ⋅ ⋅
Ϫ1 < x < 1
22.11. 22.12. 22.13. 22.14.
(1 +
x )−1/ 2
=
1−
1 2
x
+
1i 2i
3 4
x2
−
1i 2i
3i5 4i6
x3
+
⋅⋅⋅
Series for Exponential and Logarithmic Functions
22.15. 22.16. 22.17. 22.18. 22.19. 22.20.
ex
=1 +x
+
x2 2!
+
x3 3!
+ ⋅⋅⋅
ax
=
ex ln a
=1+
x ln a +
(x ln a)2 2!
+
(x ln a)3 3!
|
x
|<
π 2
0< |x| <π
|
x
|
<
π 2
0< |x| <π
|x|<1
|x|<1
22.29. 22.30.
tan−1
x
=
⎧ ⎪x
−
⎨ ⎪±
π
x3 + x5 − 35 −1+ 1
x7 7 −
+ 1
+
⎩ 2 x 3x3 5x5
cot −1
x
=
π 2
−
tan−1
x
=
⎧π
⎪⎪ ⎨
2
−
⎛ ⎝⎜x
−
x3 3
+
x5 5
−
⎩⎪⎪pπ
+
1 x
−
1 3x3
+
1 5x5
⎞ ⎠⎟
−
22.31. 22.32.
sec−1
x
=
cos−1(1/x)
=
π 2
−
⎛1 ⎝⎜ x
+
2
1 i 3x3