## 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

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