Re: Unknown Sum of Series
- To: mathgroup at smc.vnet.net
- Subject: [mg63518] Re: [mg63473] Unknown Sum of Series
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 3 Jan 2006 01:26:32 -0500 (EST)
- References: <200601021049.FAA01328@smc.vnet.net> <2CDD43D8-DF00-4EF5-B5A8-70F114D8E1CB@mimuw.edu.pl>
- Sender: owner-wri-mathgroup at wolfram.com
On 3 Jan 2006, at 10:09, Andrzej Kozlowski wrote:
>
> On 2 Jan 2006, at 19:49, Klaus G. wrote:
>
>> Mathematica 5.0 is not able to compute the symbolic sum:
>>
>> Sum[(-1)^(1 + n)*(E - ( 1 + (1/n))^n ), {n, 1, Infinity}]
>>
>> However, Nsum[...] results in 0.4456224031968407..
>>
>> I tried http://oldweb.cecm.sfu.ca/projects/ISC/ to find hidden
>> constants in that number like Pi or E, but without success.
>>
>> Any idea?
>>
>> Klaus G.
>>
>
>
> Do you have any reason to believe that there is a "closed formula"
> for this sum?
>
> It is trivial to show that this sum is convergent since this is an
> infinite alternating sum of terms whose absolute values form a
> monotonically decreasing sequence tending to zero (since Limit[( 1
> + (1/n))^n ),n->Infinity]==E). It is very easy to generate such
> sums: just take any monotonically increasing sequence of positive
> terms that tends to some limit then take the sequence of
> differences between the limit and the terms of the original
> sequence and finally take the infinite alternating sum. You will
> then get a convergent infinite sum just like the one above. In
> general there certainly will be no reason to expect any "closed
> formula" for the value of such an infinite sum. So it seems to me
> unlikely that there is any such formula here, and I suspect if
> there were one it would have been found by Ramanujan ;-)
>
> Andrzej Kozlowski
Perhaps I should have illustrated my point with an example:
Sum[(-1)^n*(1 - Cos[1/n]), {n, 1, Infinity}]
Sum[(-1)^n*(1 - Cos[1/n]), {n, 1, Infinity}]
NSum[(-1)^n*(1 - Cos[1/n]), {n, 1, Infinity}]
-0.37311820151613023
One can go on creating such examples for ever ...
Andrzej Kozlowski
- References:
- Unknown Sum of Series
- From: "Klaus G." <Karl_boehme_9@msn.com>
- Unknown Sum of Series