1
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In version 13.1 on Windows 10

DiscreteLimit[Product[Sum[1/2^k, {k, 0, i}], {i, 1, n}]/2^n,n -> Infinity]

Indeterminate

The above is not in accordance with numeric calculations

N[Table[Product[Sum[1/2^k, {k, 0, i}], {i, 1, n}]/2^n, {n, 10, 100,10}], 15]

{0.577858301626197, 0.577576465583082, 0.577576190442160, 0.577576190173467, 0.577576190173205, 0.577576190173205, 0.577576190173205, 0.577576190173205, 0.577576190173205, 0.577576190173205}

It should be noticed that 0.577576190173205 is not identified as N[EulerGamma]. Also

DiscreteMaxLimit[Product[Sum[1/2^k, {k, 0, i}], {i, 1, n}]/2^n,n -> Infinity]

is running without any response for a long time. So is Indeterminate correct?

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  • $\begingroup$ Reported as a bug/suggestion to improve. $\endgroup$ Sep 21, 2022 at 15:17

1 Answer 1

4
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In version 13.1 on Windows 10

 func = DiscreteLimit[Product[Sum[1/2^k, {k, 0, i}] // ExpandAll, {i, 1, n}]/2^n, n -> Infinity]

 (*2 QPochhammer[1/2, 1/2]*)

 N[func - EulerGamma, 100] // FunctionExpand
 (*0.00036052527167198195128735377605912913566447374144796942371489748493\
   67550360293542035386230949494068989*)

Look like is not a EulerGamma

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  • $\begingroup$ Thank you. I am out of Mathematica so I will draw a decision later. $\endgroup$
    – user64494
    Sep 20, 2022 at 14:45

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