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Math Formulas: Taylor and Maclaurin Series
Definition of Taylor series:
1.
f (x) = f (a) + f 0 (a)(x − a) +
2.
3.
Rn =
Rn =
f 00 (a)(x − a)2
f (n−1) (a)(x − a)n−1
+ ··· +
+ Rn
2!
(n − 1)!
f (n) (ξ)(x − a)n
where a ≤ ξ ≤ x,
n!
( Lagrangue’s form )
f (n) (ξ)(x − ξ)n−1 (x − a)
where a ≤ ξ ≤ x,
(n − 1)!
( Cauch’s form )
This result holds if f (x) has continuous derivatives of order n at last. If limn→+∞ Rn = 0, the infinite series
obtained is called Taylor series for f (x) about x = a. If a = 0 the series is often called a Maclaurin
series.
Binomial series
4.
n(n − 1) n−2 2 n(n − 1)(n − 2) n−3 3
(a + x)n = an + nan−1 +
a
x +
a
x + ···
2! 3!
n n−1
n n−2 2
n n−3 3
= an +
a
x+
a
x +
a
x + ···
1
2
3
Special cases of binomial series
5.
(1 + x)−1 = 1 − x + x2 − x3 + · · ·
6.
(1 + x)−2 = 1 − 2x + 3x2 − 4x3 + · · ·
−1<x<1
7.
(1 + x)−3 = 1 − 3x + 6x2 − 10x3 + · · ·
−1<x<1
8.
1·3 2 1·3·5 3
1
x −
x + ···
(1 + x)−1/2 = 1 − x +
2
2·4
2·4·6
9.
1
1 2
1·3 3
(1 + x)1/2 = 1 + x −
x +
x + ···
2
2·4
2·4·6
−1<x<1
−1<x≤1
−1<x≤1
Series for exponential and logarithmic functions
10.
11.
12.
13.
ex = 1 + x +
ex = 1 + x ln a +
x2
x3
+
+ ···
2!
3!
(x ln a)2
(x ln a)3
+
+ ···
2!
3!
x2
x3
x4
+
−
+ ··· − 1 < x ≤ 1
2
3
4
2
3
x−1
1 x−1
1 x−1
1
ln(1 + x) =
+
+
+ ··· x ≥
x
2
x
3
x
2
ln(1 + x) = x −
Series for trigonometric functions
1
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sin x = x −
14.
15.
16.
x2
x4
x6
+
−
+ ···
2!
4!
6!
22n 22n − 1 Bn x2n−1
x3
2x5
tan x = x +
+
+ ··· +
3
15
(2n)!
cos x = 1 −
17.
cot x =
18.
sec x = 1 +
19.
x3
x5
x7
+
−
+ ···
3!
5!
7!
1 x x3
22n Bn x2n−1
− −
− ··· −
x 3
45
(2n)!
−
π
π
<x<
2
2
0<x<π
x2
5x4
61x6
En x2n
π
π
+
+
+ ··· +
− <x<
2
24
720
(2n)!
2
2
2 22n − 1 En x2n
1 x 7x3
csc x = + +
+ ··· +
0<x<π
x 6
360
(2n)!
Series for inverse trigonometric functions
20.
21.
22.
23.
1 x3
1 · 3 x5
1 · 3 · 5 x7
+
+
+ ··· − 1 < x < 1
2 3
2·4 5
2·4·6 7
π
π
1 · 3 x5
1 x3
arccos x = − arcsin x = − x +
+
+ ···
−1<x<1
2
2
2 3
2·4 5

x3
x5
x7


x
−
+
−
+ ··· − 1 < x < 1


3
5
7


π
1
1
1
arctan x =
− + 3 − 5 + ··· x ≥ 1

2
x 3x
5x




 −π − 1 + 1 − 1 + · · · x < 1
2
x 3x3
5x5

π
x3
x5


− x−
+
+ ···
−1<x<1



2
3
5


π
1
1
1
arccot x = − arctan x =
− 3 + 5 − ··· x ≥ 1

2

x
3x
5x




 π + 1 − 1 + 1 − ··· x < 1
x 3x3
5x5
arcsin x = x +
Series for hyperbolic functions
24.
25.
26.
27.
sinh x = x +
x3
x5
+
+ ···
3!
5!
x2
x4
+
+ ···
2!
4!
(−1)n−1 22n 22n − 1 Bn x2n−1
x3
2x5
tanh x = x −
+
+ ···
+ ···
3
15
(2n)!
cosh x = 1 +
coth x =
(−1)n−1 22n Bn x2n−1
1 x x3
+ −
+ ···
+ ···
x 3
45
(2n)!
2
|x| <
0 < |x| < π
π
2