Re: Unknown Sum of Series
- To: mathgroup at smc.vnet.net
- Subject: [mg63517] Re: [mg63473] Unknown Sum of Series
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 3 Jan 2006 01:26:28 -0500 (EST)
- References: <200601021049.FAA01328@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
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
- References:
- Unknown Sum of Series
- From: "Klaus G." <Karl_boehme_9@msn.com>
- Unknown Sum of Series