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Re: Convergence with product

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On Wednesday, September 27, 2017 at 2:39:18 AM UTC+13, Peter Luschny wrote:
> restart;
> for n from 1 to 10 do product(1 - exp(-3*Pi*k), k=1..n): print(evalf(%,100)) od:
> product(1 - exp(-3*Pi*k), k=1..infinity);
> 
> 0.99991930048242969540076079496607565337604701848050587887793116\
>   61874620768881558525000761137393119619
> 
> 0.99991929397054310783846071160943458949489722010393675884188157\
>   72031677006591102859909094004128441835
> 
> 0.99991929397001760173579420925756466546613464451769578307221030\
>   46217577981046063251917225329379286853
> 
> 0.99991929397001755932770524384215301951312111267396474761039494\
>   20757198264951861082576558744213889585
> 
> 0.99991929397001755932428293152156543805216017938155162138450575\
>   61250456780918293082240670461486937285
> 
> 0.99991929397001755932428265534261219172472361853070175691267253\
>   46521158633329104110223861614216303137
> 
> 0.99991929397001755932428265532032468343467328512694304556439538\
>   78114792276826418913831611248576903856
> 
> 0.99991929397001755932428265532032288484350643390064827565514682\
>   19527120690353372085351014843732960367
> 
> 0.99991929397001755932428265532032288469836099442954308820325275\
>   36799279287625628587601960477165670312
> 
> 0.99991929397001755932428265532032288469834928126260024016748022\
>   44320939278639343076761038920987656959
> 
>                                0
>                                ?

With Maple 2017 (X86 64 LINUX) I got a different result:

> product(1 - exp(-3*Pi*k), k=1..infinity); 
                          infinity
                          --------'
                         '  |  |
                            |  |    (1 - exp(-3 Pi k))
                            |  |
                            |  |
                           k = 1

> evalf(product(1 - exp(-3*Pi*k), k=1..infinity),100); 
0.9999192939700175593242826553203228846983492803172770315323192841366570017\
    065263132093348972377771038

but what surprised me was

> assume(s<1/2,s>0);                                                         
> product(1-s^k,k=1..infinity);                                                 
product:   "Cannot show that 1-s^k has no zeros on [1,infinity]"                
                              infinity                                          
                              --------'                                         
                             '  |  |           k                                
                                |  |    (1 - s~ )                               
                                |  |                                            
                                |  |                                            
                               k = 1                

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Convergence with product Peter Luschny <peter.luschny@gmail.com> - 2017-09-26 06:39 -0700
  Re: Convergence with product jfh <john.harper@vuw.ac.nz> - 2017-09-26 15:50 -0700
    Re: Convergence with product Peter Luschny <peter.luschny@gmail.com> - 2017-09-30 05:42 -0700

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