Re: A paradoxical series?




On 26/06/2005 17:37, Jean-Claude Arbaut wrote:

>
>
>
> On 26/06/2005 17:30, Ann Chance wrote:
>
>> Below is a simple paradox based on a discussion in the excellent book
>> "The Road To Reality" by Sir Roger Penrose:
>>
>> Consider the fact that 1+x^2+x^4+x^6+... = (1-x^2)^(-1)
>>
>> Putting 2 into this series gives us:
>>
>> 1+2^2+2^4+2^6+... = (1-2^2)^(-1) = -1/3
>>
>> While the LHS is obviously positive, the RHS is negative. How is that
>> possible?
>>
>
> Simply because your series converges only for |x| < 1 (and maybe some |x|=1,
> but it's more difficult to study than the simple "D'Alembert criterium")).

Sorry, does not converge for |x|=1, I had sum(x^n/n) in mind ;-)

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