Harold’s Taylor Series
Cheat Sheet
20 April 2016
Power Series
Power Series About Zero
Geometric Series if 𝑎𝑛 = 𝑎
Power Series
+∞
∑ 𝑎𝑛 𝑥 𝑛 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + 𝑎4 𝑥 4 + ⋯
𝑛=0
+∞
∑ 𝑎𝑛 (𝑥 − 𝑐)𝑛 = 𝑎0 + 𝑎1 (𝑥 − 𝑐) + 𝑎2 (𝑥 − 𝑐)2 + ⋯
𝑛=0
Approximation Polynomial
𝒇(𝒙) = 𝑷𝒏 (𝒙) + 𝑹𝒏 (𝒙)
𝑃𝑛 (𝑥) = 𝑛𝑡ℎ 𝑑𝑒𝑔𝑟𝑒𝑒 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛
𝑅𝑛 (𝑥) = ± 𝐸𝑟𝑟𝑜𝑟
𝑁𝑂𝑇𝐸: 𝑃𝑛 (𝑥) 𝑖𝑠 𝑒𝑎𝑠𝑦 𝑡𝑜 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑒
Maclaurin Series
Maclaurin Series
Taylor Series centered about 𝑥 = 0
Maclaurin Series Remainder
+∞
𝒇(𝒙) ≈ 𝑷𝒏 (𝒙) = ∑
𝒏=𝟎
𝒇(𝒏) (𝟎) 𝒏
𝒙
𝒏!
𝑓 (𝑛+1) (𝑥 ∗ ) 𝑛+1
𝑥
(𝑛 + 1)!
where 𝑥 ≤ 𝑥 ∗ ≤ 𝑚𝑎𝑥 and lim 𝑅𝑛 (𝑥) = 0
𝑅𝑛 (𝑥) =
𝑥→+∞
Taylor Series
Taylor Series
Maclaurin Series if 𝑐 = 0
+∞
𝑓(𝑥) ≈ 𝑃𝑛 (𝑥) = ∑
𝑛=0
𝑓 (𝑛) (𝑐)
(𝑥 − 𝑐)𝑛
𝑛!
(𝑛+1)
Taylor Series Remainder
𝑓
(𝑥 ∗ )
(𝑥 − 𝑐)𝑛+1
(𝑛 + 1)!
where 𝑥 ≤ 𝑥 ∗ ≤ 𝑐 and lim 𝑅𝑛 (𝑥) = 0
𝑅𝑛 (𝑥) =
𝑥→+∞
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
1
Series Examples
Exponential Functions
∞
𝒙
𝒆 =∑
𝒙𝒏
𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙
𝒏!
𝒏=𝟎
∞
𝑎 𝑥 = 𝑒 𝑥 ln(𝑎) = ∑
𝑛=0
1+𝑥+
(𝑥 ln(𝑎))𝑛
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥
𝑛!
𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8
+ + + + + + +⋯
2! 3! 4! 5! 6! 7! 8!
1 + 𝑥 ln(𝑎) +
[𝑥 ln(𝑎)]2 [𝑥 ln(𝑎)]3
+
+⋯
2!
3!
Natural Logarithms
∞
ln (1 − 𝑥) = ∑
∞
ln (𝑥) = ∑
(−1)
𝑛=1
∞
𝐥𝐧 (𝟏 + 𝒙) = ∑
𝒏=𝟏
∞
𝑛=1
𝑛−1
𝑥𝑛
𝑓𝑜𝑟 |𝑥| < 1
𝑛
𝑥+
(𝑥 − 1)𝑛
𝑓𝑜𝑟 |𝑥| < 1
𝑛
𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8
+ + + + + + +⋯
2
3
4
5
6
7
8
(𝑥 − 1) −
(−𝟏)𝒏−𝟏 𝒏
𝒙 𝒇𝒐𝒓 |𝒙| < 𝟏
𝒏
𝑥−
1+𝑥
2
)=∑
ln (
𝑥 2𝑛−1 𝑓𝑜𝑟 |𝑥| < 1
1−𝑥
2𝑛 − 1
2𝑥 −
𝑛=1
(𝑥 − 1)2 (𝑥 − 1)3 (𝑥 − 1)4
+
−
+⋯
2
3
4
𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8
+ − + − + − +⋯
2
3
4
5
6
7
8
2𝑥 2 2𝑥 3 2𝑥 4 2𝑥 5 2𝑥 6 2𝑥 7
+
−
+
−
+
−⋯
3
5
7
9
11
13
Geometric Series
∞
1
= ∑(−1)𝑛 (𝑥 − 1)𝑛 𝑓𝑜𝑟 0 < 𝑥 < 2
𝑥
𝑛=0
1 − (𝑥 − 1) + (𝑥 − 1)2 − (𝑥 − 1)3 + (𝑥 − 1)4 + ⋯
∞
1
= ∑(−1)𝑛 𝑥 𝑛 𝑓𝑜𝑟 |𝑥| < 1
1+𝑥
𝑛=0
1 − 𝑥 + 𝑥2 − 𝑥3 + 𝑥4 − 𝑥5 + 𝑥6 − 𝑥7 + 𝑥8 − ⋯
∞
𝟏
= ∑ 𝒙𝒏 𝒇𝒐𝒓 |𝒙| < 𝟏
𝟏−𝒙
∞
1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + ⋯
𝒏=𝟎
1
= ∑(−1)𝑛 𝑥 2𝑛 𝑓𝑜𝑟 |𝑥| < 1
1 + 𝑥2
𝑛=0
1 − 𝑥 2 + 𝑥 4 − 𝑥 6 + 𝑥 8 − 𝑥 10 + 𝑥 12 − 𝑥 14 + ⋯
∞
1
= ∑ 𝑥 2𝑛 𝑓𝑜𝑟 |𝑥| < 1
1 − 𝑥2
1 + 𝑥 2 + 𝑥 4 + 𝑥 6 + 𝑥 8 + 𝑥 10 + 𝑥 12 + 𝑥 14 + ⋯
𝑛=0
∞
1
= ∑(−1)𝑛−1 𝑛𝑥 𝑛−1 𝑓𝑜𝑟 |𝑥| < 1
(1 + 𝑥)2
𝑛=1
1 − 2𝑥 + 3𝑥 2 − 4𝑥 3 + 5𝑥 4 − 6𝑥 5 + 7𝑥 6 − ⋯
∞
1
= ∑ 𝑛𝑥 𝑛−1 𝑓𝑜𝑟 |𝑥| < 1
(1 − 𝑥)2
1 + 2𝑥 + 3𝑥 2 + 4𝑥 3 + 5𝑥 4 + 6𝑥 5 + 7𝑥 6 + ⋯
𝑛=1
∞
(−1)𝑛−1 (𝑛 − 1)𝑛 𝑛−2
1
=
∑
𝑥
(1 + 𝑥)3
2
𝑛=2
1 − 3𝑥 + 6𝑥 2 − 10𝑥 3 + 15𝑥 4 − 21𝑥 5 + 28𝑥 6 − ⋯
𝑓𝑜𝑟 |𝑥| < 1
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
2
∞
1
(𝑛 − 1)𝑛 𝑛−2
=∑
𝑥
𝑓𝑜𝑟 |𝑥| < 1
3
(1 − 𝑥)
2
𝑛=2
∞
(−1)𝑛 (2𝑛)!
𝑥𝑛
4𝑛 (𝑛!)2 (1 − 2𝑛)
√1 + 𝑥 = ∑
𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
∞
(2𝑛)!
𝑥𝑛
√1 − 𝑥 = ∑ 𝑛
4 (𝑛!)2 (1 − 2𝑛)
𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
∞
(−1)𝑛 (2𝑛)!
2
√1 + 𝑥 = ∑ 𝑛
𝑥 2𝑛
4 (𝑛!)2 (1 − 2𝑛)
𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
∞
(2𝑛)!
2
√1 − 𝑥 = ∑ 𝑛
𝑥 2𝑛
4 (𝑛!)2 (1 − 2𝑛)
𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
1 + 3𝑥 + 6𝑥 2 + 10𝑥 3 + 15𝑥 4 + 21𝑥 5 + 28𝑥 6 + ⋯
1
1
1
5 4
7 5
21 5
1 + 𝑥 − 𝑥2 + 𝑥3 −
𝑥 +
𝑥 −
𝑥
2
8
16
128
256
1,024
+⋯
1
1
1
5 4
7 5
21 5
1 − 𝑥 − 𝑥2 − 𝑥3 −
𝑥 −
𝑥 −
𝑥
2
8
16
128
256
1,024
−⋯
1
1
1
5 8
7 10
1 + 𝑥2 − 𝑥4 + 𝑥6 −
𝑥 +
𝑥 −⋯
2
8
16
128
256
1
1
1
5 8
7 10
1 − 𝑥2 − 𝑥4 − 𝑥6 −
𝑥 −
𝑥 −⋯
2
8
16
128
256
(𝑛)‼ = 𝑛(𝑛 − 2)(𝑛 − 4) … 6 ∙ 4 ∙ 2 𝑖𝑓 𝑒𝑣𝑒𝑛
(𝑛)‼ = 𝑛(𝑛 − 2)(𝑛 − 4) … 5 ∙ 3 ∙ 1 𝑖𝑓 𝑜𝑑𝑑
𝑤ℎ𝑒𝑟𝑒 0‼ = 1 𝑎𝑛𝑑 ─1‼ = 1
Double Factorial (!!)
∞
1
√1 + 𝑥
=∑
𝑛=0
(−1)𝑛 (2𝑛 − 1)‼ 𝑛
𝑥
(2𝑛)‼
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
∞
(2𝑛 − 1)‼ 𝑛
1
=∑
𝑥
(2𝑛)‼
√1 − 𝑥
𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
∞
(−1)𝑛 (2𝑛 − 1)‼ 2𝑛
1
=∑
𝑥
(2𝑛)‼
√1 + 𝑥 2 𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
∞
(2𝑛 − 1)‼ 2𝑛
1
=∑
𝑥
(2𝑛)‼
√1 − 𝑥 2
𝑛=0
𝑓𝑜𝑟 − 1 < 𝑥 ≤ 1
1
1∙3 2 1∙3∙5 3 1∙3∙5∙7 4
1− 𝑥+
𝑥 −
𝑥 +
𝑥 −⋯
2
2∙4
2∙4∙6
2∙4∙6∙8
1
1∙3 2 1∙3∙5 3 1∙3∙5∙7 4
1+ 𝑥+
𝑥 +
𝑥 +
𝑥 +⋯
2
2∙4
2∙4∙6
2∙4∙6∙8
1
1∙3 4 1∙3∙5 6 1∙3∙5∙7 8
1 − 𝑥2 +
𝑥 −
𝑥 +
𝑥 −⋯
2
2∙4
2∙4∙6
2∙4∙6∙8
1
1∙3 4 1∙3∙5 6 1∙3∙5∙7 8
1 + 𝑥2 +
𝑥 +
𝑥 +
𝑥 +⋯
2
2∙4
2∙4∙6
2∙4∙6∙8
Binomial Series
+∞
𝑟
(1 + 𝑥) = ∑ ( ) 𝑥 𝑛
𝑛
𝑟
𝑛=0
𝑓𝑜𝑟 |𝑥| < 1 and all complex r where
𝑛
𝑟
𝑟−𝑘+1
( )=∏
𝑛
𝑘
1 + 𝑟𝑥 +
𝑟(𝑟 − 1) 2 𝑟(𝑟 − 1)(𝑟 − 2) 3
𝑥 +
𝑥 +⋯
2!
3!
𝑘=1
𝑟(𝑟 − 1)(𝑟 − 2) … (𝑟 − 𝑛 + 1)
=
𝑛!
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
3
Trigonometric Functions
∞
𝐬𝐢𝐧 (𝒙) = ∑
𝒏=𝟎
(−𝟏)𝒏
𝒙𝟐𝒏+𝟏 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙
(𝟐𝒏 + 𝟏)!
1−
𝑥 2 𝑥 4 𝑥 6 𝑥 8 𝑥 10 𝑥 12 𝑥 14
+ − + −
+
−
+⋯
2! 4! 6! 8! 10! 12! 14!
𝒏=𝟎
𝑛−1
22𝑛 (22𝑛 − 1) 𝐵2𝑛 2𝑛−1
𝑥
(2𝑛)!
𝑛=1
𝜋
𝑓𝑜𝑟 |𝑥| <
2
∞
(−1)𝑛 𝐸2𝑛 2𝑛
sec (𝑥) = ∑
𝑥
(2𝑛)!
𝑛=0
𝜋
𝑓𝑜𝑟 |𝑥| <
2
∞
(−1)𝑛−1 2 (22𝑛−1 − 1) 𝐵2𝑛 2𝑛−1
csc (𝑥) = ∑
𝑥
(2𝑛)!
tan (𝑥) = ∑
𝑥 3 𝑥 5 𝑥 7 𝑥 9 𝑥 11 𝑥 13 𝑥 15
+ − + −
+
−
+⋯
3! 5! 7! 9! 11! 13! 15!
∞
(−𝟏)𝒏 𝟐𝒏
𝐜𝐨𝐬 (𝒙) = ∑
𝒙
𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙
(𝟐𝒏)!
∞
𝑥−
1
2
17 7
62 9
𝑥 + 𝑥3 + 𝑥5 +
𝑥 +
𝑥
3
15
315
2,835
1,382 11
21,844 13
+
𝑥 +
𝑥 +⋯
155,925
608,1075
(−1)
1+
1 1
7 3
31
127
+ 𝑥+
𝑥 +
𝑥5 +
𝑥7 + ⋯
𝑥 6
360
15,120
604,800
𝑛=0
𝑓𝑜𝑟 0 < 𝑥 < 𝜋
∞
cot (𝑥) = ∑
𝑛=0
(−1)𝑛 22𝑛 𝐵2𝑛 2𝑛−1
𝑥
(2𝑛)!
𝑓𝑜𝑟 0 < 𝑥 < 𝜋
𝑥2
𝑥4
𝑥6
𝑥8
𝑥 10
+ 5 + 61 + 1,385 + 50,521
+⋯
2!
4!
6!
8!
10!
1 1
1
2 5
1
4
− 𝑥 − 𝑥3 −
𝑥 −
𝑥7 −
𝑥9 − ⋯
𝑥 3
45
189
4,725
2,835
Inverse Trigonometric Functions
∞
(2𝑛)!
−1
sin (𝑥) = ∑ 𝑛 2
𝑥 2𝑛+1
(2 𝑛!) (2𝑛 + 1)
𝑛=0
𝑓𝑜𝑟 |𝑥| ≤ 1
1
𝛤 (𝑛 + )
2
sin (𝑥) = ∑
𝑥 2𝑛+1
(2𝑛 + 1) 𝑛!
𝜋
√
𝑛=0
𝜋
−1
cos (𝑥) = − sin−1 (𝑥)
2
𝑓𝑜𝑟 |𝑥| ≤ 1
∞
𝑥+
𝑥3
1 ∙ 3𝑥 5 1 ∙ 3 ∙ 5𝑥 7 1 ∙ 3 ∙ 5 ∙ 7𝑥 9
+
+
+
+⋯
2∙3 2∙4∙5 2∙4∙6∙7 2∙4∙6∙8∙9
−1
∞
−𝟏
𝐭𝐚𝐧
(𝒙) = ∑
𝒏=𝟎
(−𝟏)𝒏 𝟐𝒏+𝟏
𝒙
(𝟐𝒏 + 𝟏)
𝒇𝒐𝒓 |𝒙| < 𝟏
𝜋
𝑥3
1 ∙ 3𝑥 5 1 ∙ 3 ∙ 5𝑥 7
−𝑥−
−
−
−⋯
2
2∙3 2∙4∙5 2∙4∙6∙7
𝑥3 𝑥5 𝑥7 𝑥9
+ − + − ⋯ 𝑓𝑜𝑟 − 1 < 𝑥 < 1
3
5
7
9
𝜋 1
1
1
1
1
− + 3 − 5 + 7 − 9 + ⋯ 𝑓𝑜𝑟 𝑥 ≥ 1
2 𝑥 3𝑥
5𝑥
7𝑥
9𝑥
𝜋 1
1
1
1
1
− − + 3 − 5 + 7 − 9 + ⋯ 𝑓𝑜𝑟 𝑥 < 1
2 𝑥 3𝑥
5𝑥
7𝑥
9𝑥
sec −1 (𝑥) = −𝑖 ln(𝑥) + 𝑖 ln(2)
∞
(2𝑛 + 1)! 𝑥 2𝑛+2
𝑖
− ∑ 𝑛
4
4 [(𝑛 + 1)!]2
𝑥 −
𝑖
3𝑖 4 5𝑖 6
−𝑖 ln(𝑥) + 𝑖 ln(2) − 𝑥 2 −
𝑥 − 𝑥 −⋯
4
32
96
𝑛=0
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
4
𝜋
2
∞
(2𝑛 + 1)! 𝑥 2𝑛+2
𝑖
+ ∑ 𝑛
4
4 [(𝑛 + 1)!]2
csc −1 (𝑥) = 𝑖 ln(𝑥) − 𝑖 ln(2) +
𝑖 ln(𝑥) − 𝑖 ln(2) +
𝜋 𝑖 2 3𝑖 4 5𝑖 6
+ 𝑥 +
𝑥 + 𝑥 +⋯
2 4
32
96
𝑛=0
𝜋
− tan−1 (𝑥)
2
𝑓𝑜𝑟 |𝑥| < 1
cot −1 (𝑥) =
Hyperbolic Functions
𝜋
𝑥3 𝑥5 𝑥7 𝑥9
− 𝑥 + − + − + ⋯ 𝑓𝑜𝑟 − 1 < 𝑥 < 1
2
3
5
7
9
1
1
1
1
1
− 3 + 5 − 7 + 9 − ⋯ 𝑓𝑜𝑟 𝑥 ≥ 1
𝑥 3𝑥
5𝑥
7𝑥
9𝑥
1
1
1
1
1
𝜋 + − 3 + 5 − 7 + 9 − ⋯ 𝑓𝑜𝑟 𝑥 < 1
𝑥 3𝑥
5𝑥
7𝑥
9𝑥
∞
𝑒 𝑥 − 𝑒 −𝑥
𝑥 2𝑛+1
sinh (𝑥) =
=∑
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥
(2𝑛 + 1)!
2
𝑥
cosh (𝑥) =
𝑒 +𝑒
2
−𝑥
𝑛=0
∞
=∑
𝑛=0
𝑥 2𝑛
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥
(2𝑛)!
𝑒 𝑥 − 𝑒 −𝑥
tanh (𝑥) = 𝑥
𝑒 + 𝑒 −𝑥
∞
2𝑛 2𝑛
2 (2 − 1) 𝐵2𝑛 2𝑛−1
tanh (𝑥) = ∑
𝑥
(2𝑛)!
𝑛=1
𝜋
𝑓𝑜𝑟 |𝑥| <
2
∞
𝐸2𝑛 2𝑛
sech (𝑥) = ∑
𝑥
(2𝑛)!
𝑛=0
𝜋
|𝑥|
𝑓𝑜𝑟
<
2
∞
(2 − 22𝑛 ) 𝐵2𝑛 2𝑛−1
1
csch (𝑥) = + ∑
𝑥
(2𝑛)!
𝑥
𝑥+
𝑥 3 𝑥 5 𝑥 7 𝑥 9 𝑥 11 𝑥 13 𝑥 15
+ + + +
+
+
+⋯
3! 5! 7! 9! 11! 13! 15!
1+
𝑥 2 𝑥 4 𝑥 6 𝑥 8 𝑥 10 𝑥 12 𝑥 14
+ + + +
+
+
+⋯
2! 4! 6! 8! 10! 12! 14!
𝑥3
𝑥5
𝑥7
𝑥9
𝑥 11
+ 16 − 272 + 7,936 − 353,792
3!
5!
7!
9!
11!
+⋯
1
2
17 7
62 9
1,382 11
𝑥 − 𝑥3 + 𝑥5 −
𝑥 +
𝑥 −
𝑥
3
15
315
2,835
155,925
+⋯
𝑥−2
𝑥2
𝑥4
𝑥6
𝑥8
𝑥 10
1 − + 5 − 61 + 1,385 − 50,521
+⋯
2!
4!
6!
8!
10!
1 1
7 3
31
127
− 𝑥+
𝑥 −
𝑥5 +
𝑥7 − ⋯
𝑥 6
360
15,120
604,800
𝑛=1
𝑓𝑜𝑟 0 < |𝑥| < 𝜋
∞
1
(−1)𝑛−1 22𝑛 𝐵2𝑛 2𝑛−1
coth (𝑥) = + ∑
𝑥
(2𝑛)!
𝑥
𝑛=1
𝑓𝑜𝑟 0 < |𝑥| < 𝜋
1 1
1
2 5
1
4
+ 𝑥 − 𝑥3 +
𝑥 −
𝑥7 +
𝑥9 − ⋯
𝑥 3
45
945
4,725
2,835
Inverse Hyperbolic Functions
∞
−1
sinh (𝑥) = ∑
𝑛=0
(−1)𝑛 (2𝑛)!
𝑥 2𝑛+1
(2𝑛 𝑛!)2 (2𝑛 + 1)
𝑓𝑜𝑟 |𝑥| ≤ 1
∞
(−1)𝑛 (2𝑛 − 1)‼ 2𝑛+1
sinh−1 (𝑥) = ∑
𝑥
(2𝑛 + 1)(2𝑛)‼
𝑥−
𝑥3
1 ∙ 3𝑥 5 1 ∙ 3 ∙ 5𝑥 7 1 ∙ 3 ∙ 5 ∙ 7𝑥 9
+
−
+
−⋯
2∙3 2∙4∙5 2∙4∙6∙7 2∙4∙6∙8∙9
𝑛=0
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
5
∞
𝜋
2−𝑛
cosh (𝑥) = 𝑖 − 𝑖 ∑
𝑥 2𝑛+1
2
𝑛! (2𝑛 + 1)
−1
𝜋𝑖
𝑖 𝑥 3 𝑖 ∙ 1 ∙ 3𝑥 5 𝑖 ∙ 1 ∙ 3 ∙ 5𝑥 7
−𝑖𝑥−
−
−
−⋯
2
2∙3
2∙4∙5
2∙4∙6∙7
𝑛=0
𝑓𝑜𝑟 |𝑥| ≤ 1
∞
𝑥 2𝑛+1
−1
tanh (𝑥) = ∑
2𝑛 + 1
𝑥+
𝑛=0
𝑥 3 𝑥 5 𝑥 7 𝑥 9 𝑥 11 𝑥 13 𝑥 15
+ + + +
+
+
+⋯
3
5
7
9
11
13
15
𝑓𝑜𝑟 |𝑥| < 1, 𝑥 ≠ ±1
−1
sech (𝑥) = − ln(𝑥) + ln(2)
∞
(2𝑛 + 1)! 𝑥 2𝑛+2
1
+ ∑ 𝑛
4
4 [(𝑛 + 1)!]2
1
3𝑖
5𝑖
− ln(𝑥) + ln(2) + 𝑥 2 + 𝑥 4 + 𝑥 6 + ⋯
4
32
96
csch−1 (𝑥) = − ln(𝑥) + ln(2)
∞
1
(−1)𝑛 (2𝑛 + 1)! 𝑥 2𝑛+2
+ ∑
4
4𝑛 [(𝑛 + 1)!]2
1
3𝑖
5𝑖
− ln(𝑥) + ln(2) + 𝑥 2 − 𝑥 4 + 𝑥 6 − ⋯
4
32
96
𝑛=0
𝑛=0
∞
𝑖𝜋
𝑥 2𝑛+1
coth (𝑥) = − + ∑
2
2𝑛 + 1
−1
=−
𝑛=0
Bernoulli Numbers
𝐵0 = 1
1
𝐵1 = −
2
1
𝐵2 =
6
1
𝐵4 = −
30
1
𝐵6 =
42
1
𝐵8 = −
30
5
𝐵10 =
66
691
𝐵12 = −
2,730
7
𝐵14 =
6
3,617
𝐵16 = −
510
438,675
𝐵18 =
798
174,611
𝐵20 = −
330
854,513
𝐵22 =
138
𝑖𝜋
𝑥 3 𝑥 5 𝑥 7 𝑥 9 𝑥 11 𝑥 13
+𝑥+ + + + +
+
+⋯
2
3
5
7
9
11
13
Euler Numbers
𝐸0 = 1
𝐸1 = 0
𝐸2 = −1
𝐸3 = 0
𝐸4 = 5
𝐸5 = 0
𝐸6 = −61
𝐸8 = 1,385
𝐸10 = −50,521
𝐸12 = 2,702,765
𝐸14 = −199,360,981
𝐸16 = 19,391,512,145
𝐸18 = −2,404,879,675,441
𝐸20 = 370,371,188,237,525
𝐸22 = −69,348,874,393,137,901
𝐸2𝑛+1 = 0
Gamma Function
1
𝛤0 = 𝛤 ( ) = √𝜋
2
3
√𝜋
𝛤1 = 𝛤 ( ) =
2
2
5
3√𝜋
𝛤2 = 𝛤 ( ) =
2
4
7
15√𝜋
𝛤3 = 𝛤 ( ) =
2
8
9
105√𝜋
𝛤4 = 𝛤 ( ) =
,
2
16
11
945√𝜋
𝛤5 = 𝛤 ( ) =
2
32
13
10,395√𝜋
𝛤6 = 𝛤 ( ) =
2
64
15
135,135√𝜋
𝛤7 = 𝛤 ( ) =
2
128
17
2,027,025√𝜋
𝛤8 = 𝛤 ( ) =
2
256
19
34,459,425√𝜋
𝛤9 = 𝛤 ( ) =
2
512
21
654,729,075√𝜋
𝛤10 = 𝛤 ( ) =
2
1,024
Generating Function
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
6
∞
𝛤(𝑡) = ∫ 𝑥 𝑡−1 𝑒 −𝑥 𝑑𝑥
∞
𝑡
𝐵𝑛 𝑛
=∑
𝑡
𝑡
𝑒 −1
𝑛!
𝑛=0
0
∞
1
2
𝐸𝑛 𝑛
= 𝑡
=∑
𝑡
−𝑡
cosh(𝑡) 𝑒 + 𝑒
𝑛!
𝑛=0
Recursive Definition
𝐵𝑚 (𝑛)
Iterated Sum
𝑚−1
𝐸2𝑛
𝑘=0
𝑘 (−1)𝑗 (𝑘 − 2𝑗)2𝑛+1
= 𝑖 ∑ ∑( )
𝑗
2𝑘 𝑖 𝑘 𝑘
𝑚
𝐵𝑘 (𝑛)
= 𝑛𝑚 − ∑ ( )
𝑘 𝑚−𝑘+1
𝐵0 (𝑛) = 1
2𝑛+1
𝑘
𝑘=1 𝑗=0
(2𝑛)!
1
𝛤 ( + 𝑛) = 𝑛 √𝜋
2
4 𝑛!
(𝑛 − 2)‼
𝑛
𝛤( ) =
𝑛−1 √𝜋
2
2 2
Recursive Definition
𝛤(𝑛 + 1) = 𝑛 ∙ 𝛤(𝑛)
𝑛
𝑛−2
𝑛−2
)∙𝛤(
)
𝛤( ) = (
2
2
2
1
𝛤 ( ) = √𝜋
2
References:
https://www.wolframalpha.com
https://en.wikipedia.org
http://ddmf.msr-inria.inria.fr/1.9.1/ddmf
http://web.mit.edu/kenta/www/three/taylor.html
Copyright © 2015-2016 by Harold Toomey, WyzAnt Tutor
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