User:Marco Polo

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[edit] Friends

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[edit] Me

Wikipedia - Marco Polo (talk | contribs)

Wikisource - Marco Polo (talk | contribs)

Wookieepedia - Darth Gama (talk | contribs)

[edit] Destroyer 45

Wikipedia - Destroyer 45 (talk | contribs)

Wookieepedia - Star Destroyer 2500 (talk | contribs)

[edit] Esperanza

[edit] A Proof that π really is irrational

Theorem: π is irrational

Proof: Suppose \pi=\frac{p}{q}, where p and q are integers. Consider the functions f_n(x)\! defined on [0, \pi]\! by

f_n(x)=\frac{q^nx^n(\pi-x)^n}{n!} = \frac{x^n(p-qx)^n}{n!}

Clearly f_n(0) = f_n(\pi) = 0\! for all n. Let f_n^{(m)}(x) denote the m-th derivative of f_n(x)\!. Note that

f_n^{(m)}(0) = - f_n^{(m)}(\pi) = 0\mbox{ for }m <= n\mbox{ or for }m > 2n; otherwise some integer

\operatorname{max}\ f_n(x) = f_n\left(\frac{\pi}{2}\right) = \frac{q^n\left(\frac{\pi}{2}\right)^{2n}}{n!}

By repeatedly applying integration by parts, the definite integrals of the functions f_n(x) \sin x\! can be seen to have integer values. But f_n(x) \sin x\! are strictly positive, except for the two points 0 and π, and these functions are bounded above by \frac{1}{\pi} for all sufficiently large n. Thus for a large value of n, the definite integral of f_n(x) \sin x\! is some value strictly between 0 and 1, a contradiction.

[edit] Userboxes

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