842 APPENDIX D
D.4 SERIES EXPANSIONS AND FINITE SERIES
( 1 ±x)n= 1 ±nx+n(n− 1 )
2!x^2 ±n(n− 1 )(n− 2 )
3!x^3 ±... (x^2 < 1 )Binomial( 1 ±x)−n= 1 ∓nx+n(n− 1 )
2!x^2 ∓n(n+ 1 )(n+ 2 )
3!x^3 ∓... (x^2 < 1 )Binomialex= 1 +x+1
2!x^2 +... (all real values ofx)Exponentialsinx=x−1
3!x^3 +1
5!x^5 −... (all real values ofx)Trigonometriccosx= 1 −1
2!x^2 +1
4!x^4 −... (all real values ofx)Trigonometricf(x)=f(a)+(x−a)f′(a)+(x−a)^2
2!f′′(a)+(x−a)^3
3!f′′′(a)+...Taylorf(x+h)=f(x)+hf′(x)+h^2
2!f′′(x)+h^3
3!f′′′(x)+...=f (h)+xf′(h)+x^2
2!f′′(h)+x^3
3!f′′′(h)+...TaylorFinite Series:
∑Nn= 1n=N(N+ 1 )
2
∑Nn= 1n^2 =N(N+ 1 )( 2 N+ 1 )
6
∑Nn= 1n^3 =N^2 (N+ 1 )^2
4
∑Nn= 0xn=xN+^1 − 1
x− 1
∑Nn= 0ej(θ+nφ)=sin [(N+ 1 )φ/2]
sin(φ/ 2 )ej[θ+(N φ/^2 )]