Monday 15 September 2008

Problems on Factorization

Level1: Factorize the following (OR expand then factorize.) (2x-y)^2-(x+2y)^2 81a^4-256 (x-4)^2-18x Level2: Factorize the following. (OR expand then factorize) (a^2)(1+2b)+(b^2)(1+2a)+ab(2+ab) (a^2)(b+c)+(b^2)(a+c)+(c^2)(a+b)+2abc x^2+4x+xy+3y+3 (that seems easy.) Level X(?) List all pair (ax+b)(cx+d)=12x^2+ex+35(-1)^f that a-f is integer. Prove or disprove exist pair (a,c) that ax^2+bx+c can be factorize as (ex+/-f)(gx+/-h) While changing b. and a-g are a)Integers? b)Reals? c)Complex number?

Thursday 11 September 2008

About harmonic series.

*JSL* Harmonic series is a special sequence that touch infinitive but very slow. There's so many ways to prove it. Let harmonic series be S and it x th partial sum if S(s). S=1+1/2+... =1 + (1/2) + (1/3+1/4) +(1/5+1/6+1/7+1/8)...←→S(1)+(S(2)-S(1))+(S(4)-S(2))...+(S(2^p)-S(s^(p-1))... >1+1/2+1/2+1/2....... => infinite. *JSL+* The let of harmoninc series's notation is still exist. S=(1+1/3+1/5...)+(1/2+1/4+1/6...) =(1+1/3+1/5...)+0.5S >(1/2+1/4+1/6...)+0.5S+0.5 =S+0.5 S>S+0.5 only exist when S is infinite. *SSL* The sum of prime^-1 is also infinite. Prove. Notation: P is the sum and P(x) is partial sum. S is the sum of harmonic series. ln S = Sum of (Prime)ln 1/(1-p^-1)= sum of (prime) -(ln (1-p^-1)) ln S = infinite