Series Formulas



Series Formulas

1. Arithmetic and Geometric Series 2. Special Power Series

Definitions:

Powers of Natural Numbers

First term: a1

Nth term: an

Number of terms in the series: n

Sum of the first n terms: Sn

Difference between successive terms: d

Common ratio: q

Sum to infinity: S

n

k =1

n

��k

an = a1 + ( n ? 1) d

2a1 + ( n ? 1) d

2

=

1

n ( n + 1)( 2n + 1)

6

3

=

1 2

2

n ( n + 1)

4

n

��k

Special Power Series

1

= 1 + x + x 2 + x3 + . . .

1? x

1

= 1 ? x + x2 ? x3 + . . .

1+ x

a + ai +1

ai = i ?1

2

a + an

Sn = 1

?n

2

Sn =

2

k =1

k =1

Arithmetic Series Formulas:

1

�� k = 2 n ( n + 1)

?n

an = a1 ? q n?1

ln (1+ x ) = x ?

Sn =

S=

an q ? a1

q ?1

(

n

x3 x5 x 7 x9

+ ? +

...

3! 5! 7! 9!

cos x = 1 ?

x 2 x 4 x6 x8

+ ? +

...

2! 4! 6! 8!

tan x = x +

)

a1 q ? 1

q ?1

a1

1? q

x2 x3 x 4 x5

+ ? + ...

2 3 4 5

sin x = x ?

ai = ai ?1 ? ai +1

Sn =

( for : ? 1 < x < 1)

x 2 x3

+ + ...

2! 3!

ex = 1 + x +

Geometric Series Formulas:

( for : ? 1 < x < 1)

x3 2x5 17x7

+

+

+ ...

3 15 315

sinh x = x +

x3 x5 x 7 x 9

+ + +

...

3! 5! 7! 9!

cosh x = 1 +

x 2 x 4 x6 x8

+ + +

...

2! 4! 6! 8!

for ? 1 < q < 1

tan x = x ?

x3 2x5 17x7

+

?

+...

3 15 315

( for : ?1 < x < 1)

��

��?

?

? for : ? < x < ?

2

2?

?

��

��?

?

? for : ? < x < ?

2

2?

?



3. Taylor and Maclaurin Series

Definition:

f(

f ���(a )( x ? a ) 2

+ . . .+

2!

f ( x) = f (a ) + f ��(a ) ( x ? a ) +

Rn =

Rn =

f(

n)

(�� )( x ? a )

n)

(a) ( x ? a )

n ?1

( n ? 1)!

+ Rn

n

n!

f(

n ?1)

Lagrange ' s form

a �ܦ� �� x

Cauch ' s form

a �ܦ� �� x

n ?1

(�� )( x ? �� ) ( x ? a )

( 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

(a + x)

n

= a n + na n?1 x +

n ( n ? 1)

a n?2 x 2 +

n ( n ? 1)( n ? 2 )

2!

3!

?n?

?n?

?n?

= a n + ? ? a n ?1 x + ? ? a n? 2 x 2 + ? ? a n ?3 x3 + ...

?1?

? 2?

?3?

a n? 3 x3 + ...

Special cases:

(1 + x )

?1

= 1 ? x + x 2 ? x3 + x 4 ? ...

?1 < x < 1

(1 + x )

?2

= 1 ? 2 x + 3 x 2 ? 4 x3 + 5 x 4 ? ...

?1 < x < 1

(1 + x )

?3

= 1 ? 3 x + 6 x 2 ? 10 x3 + 15 x 4 ? ...

?1 < x < 1

(1 + x )

?

1

2

= 1?

1

(1 + x ) 2

= 1+

1

1? 3 2 1 ? 3 ? 5 3

x+

x ?

x + ...

2

2?4

2?4?6

1

1 2

1? 3 3

x?

x +

x + ...

2

2?4

2?4?6

?1 < x �� 1

?1 < x �� 1

Series for exponential and logarithmic functions

ex = 1 + x +

x 2 x3

+

+ ...

2! 3!

x

a = 1 + x ln a +

ln (1 + x ) = x ?

( x ln a )

2!

2

3

+

( x ln a )

3!

+ ...

x 2 x3 x 4

+

? ...

2

3

4

2

?1 < x �� 1

3

? x ?1? 1 ? x ?1? 1 ? x ?1 ?

ln (1 + x ) = ?

?+ ?

? + ?

? + ...

? x ? 2? x ? 3? x ?

x��

1

2



Series for trigonometric functions

sin x = x ?

x3 x5 x7

+

?

+ ...

3! 5! 7!

cos x = 1 ?

x2 x4 x6

+

?

+ ...

2! 4! 6!

(

)

2 2 n 22 n ? 1 Bn x 2 n ?1

x 3 2 x5 17 x 7

tan x = x +

+

+

+ ... +

3

15

315

( 2n ) !

?

2 2 n Bn x 2 n ?1

1 x x3 2 x 5

cot x = ? ?

?

? ... ?

x 3 45 945

( 2n )!

0< x ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download