Series Formulas

1. Arithmetic and Geometric Series 2. Special Power Series

Definitions: Powers of Natural Numbers
First term: a1 n
1
Nth term: an ∑ k = 2 n ( n + 1)
k =1
Number of terms in the series: n
n
1
Sum of the first n terms: Sn
Difference between successive terms: d
∑k
k =1
2
=
6
n ( n + 1)( 2n + 1)

Common ratio: q n
1 2
∑k
2
Sum to infinity: S
3
= n ( n + 1)
k =1 4
Arithmetic Series Formulas:
Special Power Series
an = a1 + ( n − 1) d
1
a + ai +1
= 1 + x + x 2 + x3 + . . . ( for : − 1 < x < 1)
ai = i −1 1− x
2 1
a + an
= 1 − x + x2 − x3 + . . . ( for : − 1 < x < 1)
Sn = 1 ⋅n 1+ x
2 x 2 x3
ex = 1 + x + + + ...
2a1 + ( n − 1) d 2! 3!
Sn = ⋅n
2 x 2 x 3 x 4 x5
ln (1+ x ) = x − + − + ... ( for : −1 < x < 1)
Geometric Series Formulas: 2 3 4 5
an = a1 ⋅ q n−1 x3 x5 x 7 x9
sin x = x − + − + ...
3! 5! 7! 9!
ai = ai −1 ⋅ ai +1
x 2 x 4 x6 x8
an q − a1 cos x = 1 − + − + ...
Sn = 2! 4! 6! 8!
q −1 x3 2x5 17x7  π π
tan x = x + + + + ...  for : − < x < 
Sn =
( n
a1 q − 1 ) 3 15 315  2 2
q −1 x3 x5 x 7 x 9
sinh x = x + + + + ...
a1 3! 5! 7! 9!
S= for − 1 < q < 1
1− q x 2 x 4 x6 x8
cosh x = 1 + + + + ...
2! 4! 6! 8!
x3 2x5 17x7  π π
tan x = x − + − +...  for : − < x < 
3 15 315  2 2
3. Taylor and Maclaurin Series

Definition:
f(
n −1) n −1
f ′′(a )( x − a ) 2 (a) ( x − a )
f ( x) = f (a ) + f ′(a ) ( x − a ) + + . . .+ + Rn
2! ( n − 1)!
f(
n) n
(ξ )( x − a )
Rn = Lagrange ' s form a ≤ξ ≤ x
n!
f(
n) n −1
(ξ )( x − ξ ) ( x − a )
Rn = Cauch ' s form a ≤ξ ≤ x
( n − 1)!

This result holds if f(x) has continuous derivatives of order n at last. If lim Rn = 0 , the infinite series obtained is called
n →∞
Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series.

Binomial series
n n ( n − 1) n ( n − 1)( n − 2 )
(a + x) = a n + na n−1 x + a n−2 x 2 + a n− 3 x3 + ...
2! 3!
n n n
= a n +   a n −1 x +   a n− 2 x 2 +   a n −3 x3 + ...
1  2 3
Special cases:
−1
(1 + x ) = 1 − x + x 2 − x3 + x 4 − ... −1 < x < 1
−2
(1 + x ) = 1 − 2 x + 3 x 2 − 4 x3 + 5 x 4 − ... −1 < x < 1
−3
(1 + x ) = 1 − 3 x + 6 x 2 − 10 x3 + 15 x 4 − ... −1 < x < 1
1
− 1 1⋅ 3 2 1 ⋅ 3 ⋅ 5 3
(1 + x ) 2 = 1− x+ x − x + ... −1 < x ≤ 1
2 2⋅4 2⋅4⋅6
1
1 1 2 1⋅ 3 3
(1 + x ) 2 = 1+ x− x + x + ... −1 < x ≤ 1
2 2⋅4 2⋅4⋅6

Series for exponential and logarithmic functions

x 2 x3
ex = 1 + x + + + ...
2! 3!
2 3
x ( x ln a ) ( x ln a )
a = 1 + x ln a + + + ...
2! 3!
x 2 x3 x 4
ln (1 + x ) = x − + − ... −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
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Series for trigonometric functions
x3 x5 x7
sin x = x − + − + ...
3! 5! 7!
x2 x4 x6
cos x = 1 − + − + ...
2! 4! 6!

tan x = x +
x 3 2 x5 17 x 7
+ + + ... +
( )
2 2 n 22 n − 1 Bn x 2 n −1
−
π
<x<
π
3 15 315 ( 2n ) ! 2 2
1 x x3 2 x 5 2 2 n Bn x 2 n −1
cot x = − − − − ... − 0< x <π
x 3 45 945 ( 2n )!
x 2 5 x 4 61x 6 E x2n π π
sec x = 1 + + + + ... + n + ... − <x<
2 24 720 ( 2n ) ! 2 2

1 x 7 x3
csc x = + + + ... +
( )
2 2 2 n − 1 En x 2 n
+ ... 0< x <π
x 6 360 ( 2n )!
1 x 3 1 ⋅ 3 x5 1 ⋅ 3 ⋅ 5 x 7
sin −1 x = x + ⋅ + ⋅ + + ... −1 < x < 1
2 3 2⋅4 5 2⋅4⋅6 7
π π  1 x3 1 ⋅ 3 x5 
cos −1 x = − sin −1 x = −x+ ⋅ + ⋅ + ...  −1 < x < 1
2 2  2 3 2⋅4 5 
 x3 x 2 x3
x − + − + ... if − 1 < x < 1
 3 5 7
 π 1 1 1
tan − 1 x =  − + − + ... if x ≥ 1
3
 2 x 3x 5 x5
 π 1 1 1
 − − + 3 − 5 + ... if x < 1
 2 x 3x 5x
π  x3 x 2 x 3 
 −x− + − + ..  . if −1 < x < 1

2  3 5 7 

−1 π −1


cot x = − tan x =  1 1 1 1
2 − + − + ... if x ≥ 1
 x 3x 3
5x 5
7 x7
 1 1 1 1
 π+ − + − + ... if x < 1
 x 3x 3
5x 5
7 x7
Series for hyperbolic functions
x3 x5 x 7
sinh x = x + + + + ...
3! 5! 7!
x2 x4 x6
cosh x = 1 + + + + ...
2! 4! 6!

tanh x = x −
x3 2 x5 17 x7
+ − + ... +
( −1) n −1 2 n
2 ( 22n − 1) Bn x2n−1 + ... if −
π
<x<
π
3 15 315 ( 2n )! 2 2
n −1 2 n
1 x x3 2 x 7 ( −1) 2 Bn x 2n−1
coth x = + − + + ... + + ... if 0 < x < π
x 3 45 945 ( 2n ) !