Binomial Theorem

  • ( 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 )