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Topic: Sequence limit
Replies: 72   Last Post: Nov 26, 2013 12:07 AM

 Messages: [ Previous | Next ]
 Roland Franzius Posts: 586 Registered: 12/7/04
Re: Sequence limit
Posted: Oct 5, 2013 1:14 AM

Am 04.10.2013 11:50, schrieb quasi:
> Roland Franzius wrote:
>>
>> Take any sequence of rational approximations of pi
>>
>> r_i = m_k/n_k -> pi eg the continued fraction for atan 1.
>>
>> Since pi is not rational, there is no upper limit on n_k.
>>
>> The continued fractions are the best rational approximations,
>> the limit is never more than 1/n_k away.
>>
>> Take a subsequence with denominator n_(k_j) of more than
>> exponential growth eg e^(n_k_j^2).
>>
>> Then
>>
>> abs(sin n_(k_j))
>> = abs(sin ( m_(k_j) pi+ eps_j))
>> = abs(sin eps_j)
>> < 1/2 eps_j

>
> How do you justify the claim
>
> abs(sin(eps_j)) < (1/2)*eps_j
>
> ??
>

We called it the epsilon trick.

In full generality, in a logic oriented freshmans analyis course it was
once shown, that you can change epsilon by any positive factor if you
find in the course of a proof that the outcome is bigger than epsilon.

The three epsilon trick is to show that something is less than
epsilon/3, so there is some space left for an unforseen blow up by a
triangle inequality.

--

Roland Franzius

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