[seqfan] Re: 2(x - 1)^x = x^x

[aleksisto]

Let y=1/x, and so y=1+2^(-y).
Maybe y has an interesting binary expansion.

aleksisto

----- Original Message ----- 
From: "Neil Sloane" <njasloane at gmail.com>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Friday, March 02, 2018 3:30 PM
Subject: [seqfan] Re: 2(x - 1)^x = x^x


>> shall we also have the decimal expansion of the number discussed here as 
>> a
> sequence?
>
> Of course!
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Fri, Mar 2, 2018 at 7:05 AM, Hugo Pfoertner <yae9911 at gmail.com> wrote:
>
>> Given that we get hits like  "Decimal expansion of log_10(19)"
>> https://oeis.org/A155062 by a search
>> https://oeis.org/search?q=7%2C2%2C2%2C8%2C9&sort=&language=&go=Search
>> shall we also have the decimal expansion of the number discussed here as 
>> a
>> sequence?
>>
>> On Thu, Mar 1, 2018 at 3:51 PM, <rgwv at rgwv.com> wrote:
>>
>> > And that is what I got yesterday. RGWv
>> >
>> > -----Original Message-----
>> > From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of neil greubel
>> > Sent: Wednesday, 28 February, 2018 11:53 PM
>> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> > Subject: [seqfan] Re: 2(x - 1)^x = x^x
>> >
>> > The exact solution to x^x = 2*(1-x)^x is x = 1 / (1 + W(ln2/2)/ln2) ,
>> > where W(x) is the Lambert W-function.
>> >
>> > Proof: By taking the ln of both sides of the equation x lnx = ln2 + x
>> > ln(1-x) leads to
>> > - (ln2/x) = ln((1-x)/x)
>> > exp(- ln2/x) = (1-x)/x
>> > (1/2) = (1/x - 1) * exp( ln2 / x) * (1/2)
>> > (ln2 / 2) = ln2 * (1/x - 1) * exp(ln2 / x - ln2) =>
>> > ln2 *(1/x - 1) = W(ln2 / 2)
>> > 1/x = 1 + W(ln2 / 2) / ln2
>> > and finally
>> > x = ln2 / (ln2 + W(ln2/2)) = 0.72289208118314779553.....
>> >
>> > On Wed, Feb 28, 2018 at 11:17 PM, <rgwv at rgwv.com> wrote:
>> >
>> > > 0.722892081183147795532365351219771674355877736768641720125436
>> > > 5151991334348752469431309465384425833668792370029595892746086...
>> > >
>> > > -----Original Message-----
>> > > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
>> > > WALTER KEHOWSKI
>> > > Sent: Wednesday, 28 February, 2018 6:18 PM
>> > > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>; юрий
>> > > герасимов <2stepan at rambler.ru>
>> > > Subject: [seqfan] Re: 2(x - 1)^x = x^x
>> > >
>> > > The equation 2(1-x)^x = x^x has the zero x=0.7228920812. HTH.
>> > >
>> > > > On February 26, 2018 at 2:32 AM юрий герасимов <2stepan at rambler.ru>
>> > > wrote:
>> > > >
>> > > >
>> > > > Dear SeqFans.
>> > > > Help find the first members of this sequence:
>> > > > Decimal expansion of the real positive solution to 2(x - 1)^x = 
>> > > > x^x.
>> > > > Thanks. JSG.
>> > > >
>> > > > --
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>> > >
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>> >
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