User:Guardian of Light

Some pages I've invested a lot in:

1. Differentiable manifold
2. Proofs of Fermat's little theorem
3. John Derbyshire
4. Ayla (Chrono Trigger)
5. Frog (Chrono Trigger)
6. Sonic Heroes

Mathematical

1. Trigonometric function
2. Hyperbolic function
3. Taylor series
4. Taylor's theorem
5. Table of integrals
6. Derivative
7. Partial fraction
8. Metamathematics
9. Coversine
10. Critical line theorem
11. Fermat's little theorem
12. Limit (mathematics)
13. Domain (mathematics)
14. Laplace transform
15. 173 (number)
16. Amortization (business)

Scientific

1. Carl Friedrich Gauss
2. Chaos theory
3. Wave equation
4. P-branes
5. Great White Shark

Miscellaneous

1. Chrono Trigger
2. Melchior (Chrono Trigger)
3. English Language
4. Crono
5. Serge (Chrono Cross)
6. Kid (Chrono Cross)
7. Lynx (Chrono Cross)
8. Luccia
9. Harle

The pages I myself have made (from ye olde scratch) are--in chronological order:

1. Haversine
2. This Page
3. Heliotrope
4. Magnometer
5. Belthasar (Chrono Trigger)
6. 12000 B.C. (Chrono Trigger)

To show the probability that two integers chosen at random are relatively prime is ${\displaystyle {6 \over \pi ^{2}}}$.

Proof: It is sufficient to show ${\displaystyle \sum _{n=1}^{\infty }{1 \over n^{2}}={\pi ^{2} \over 6}}$. When we have a polynomial with constant term one, we may rewrite it in factored form as follows: If ${\displaystyle \alpha _{1},\alpha _{2},...,\alpha _{r}\,}$ are the roots of a polynomial p(z), then we may write ${\displaystyle p(z)=\left(1-{z \over \alpha _{1}}\right)...\left(1-{z \over \alpha _{r}}\right)}$.

Now examine the power series for the function sin(z)/z. ${\displaystyle {\sin z \over z}=1-{z^{2} \over 3!}+{z^{4} \over 5!}+...+{(-1)^{n}z^{2n} \over (2n+1)!}}$

Well we also know we can rewrite sin(z)/z in terms of its roots to be:

${\displaystyle \left(1-{z \over \pi }\right)\left(1+{z \over \pi }\right)\left(1-{z \over 2\pi }\right)\left(1+{z \over 2\pi }\right)\left(1-{z \over 3\pi }\right)\left(1+{z \over 3\pi }\right)...\left(1-{z \over k\pi }\right)\left(1+{z \over k\pi }\right)...}$

If we examine the quadratic term in each we find that:

${\displaystyle {1 \over 3!}={1 \over \pi ^{2}}\sum _{n=1}^{\infty }{1 \over n^{2}}\rightarrow {\pi ^{2} \over 6}=\sum _{n=1}^{\infty }{1 \over n^{2}}{\text{Q.E.D.}}}$