- ( 1+x )n = 1+nx+[n( n-1)/2!] .x2 + [n(n-1)(n-2)/3!].x3 +……
- ( 1+x )-n = 1-nx+[-n( n+1)/2!] .x2 – [n(n+1)(n+2)/3!].x3 +……
If x<<1 then x2,x3,…. is negligible. so:
- (1+x ) -n ≈ 1-nx
- (1-x ) n ≈ 1-nx
- (1-x ) -n ≈ 1+nx
Exponential Series
- ex = 1 + x/1! + x2 /2! + x3/3! + …..
- e = 1 + 1/1! + 1/2! + 1/3! + ….
- e = 2.7182
- e-x = 1 – x/1! + x2 /2! – x3/3! + …..
- ex + e-x = 2 [ 1 + x2/2! + x4/4 + …..]
Factors
- ( a+b )2 = a2 + b2 + 2ab
- ( a-b )2 = a2 + b2 – 2ab
- ( a2 – b2) = ( a+b ) ( a-b )
- ( a2 + b2 ) = ( a+b )2– 2ab
- ( a+b )3 = a3 + b3 + 3ab( a+b )
- ( a-b )3 = a3 – b3 – 3ab( a-b )
- ( a+b+c)2 = a2 + b2 + c2 +2(ab + bc + ac )
- a3 + b3 + c3 – 3 abc = ( a+b+c ) ( a2 + b2 + c2 – ab – bc – ac )
- ( a+b )4 = a4 + b4 + 2ab ( 2a2 + 3ab + 2b2)
- ( a-b )4 = a4 + b4 – 2ab ( 2a2 + 3ab – 2b2 )