phy1020.DVI

Appendix D

Useful Series

The first four series are valid ifjxj<1; the fifth is valid forx^2 <a^2 ; and the last three are valid for all real
x.

.1Cx/1=2D 1 C

1

2

x

1

8

x^2 C

1

16

x^3 

5

128

x^4 C

7

256

x^5 

21

1024

x^6 C

33

2048

x^7 

429

32768

x^8 C(D.1)

.1x/1=2D 1 

1

2

x

1

8

x^2 

1

16

x^3 

5

128

x^4 

7

256

x^5 

21

1024

x^6 

33

2048

x^7 

429

32768

x^8 (D.2)

.1Cx/1=2D 1 

1

2

xC

3

8

x^2 

5

16

x^3 C

35

128

x^4 

63

256

x^5 C

231

1024

x^6 

429

2048

x^7 C

6435

32768

x^8 (D.3)

.1x/1=2D 1 C

1

2

xC

3

8

x^2 C

5

16

x^3 C

35

128

x^4 C

63

256

x^5 C

231

1024

x^6 C

429

2048

x^7 C

6435

32768

x^8 C(D.4)

1

aCx

D

1

a



x

a^2

C

x^2

a^3



x^3

a^4

C

x^4

a^5



x^5

a^6

C (D.5)

exD 1 CxC

x^2

2

C

x^3

6

C

x^4

24

C

x^5

120

C

x^6

720

C

x^7

5040

C

x^8

40320

C

x^9

362880

C (D.6)

sinxDx

x^3

6

C

x^5

120



x^7

5040

C

x^9

362880



x^11

39916800

C

x^13

6227020800

 (D.7)

cosxD 1 

x^2

2

C

x^4

24



x^6

720

C

x^8

40320



x^10

3628800

C

x^12

479001600

 (D.8)