# Possibly incorrect result of DiscreteLimit

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?

• Reported as a bug/suggestion to improve. Commented Sep 21, 2022 at 15:17

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

• Thank you. I am out of Mathematica so I will draw a decision later. Commented Sep 20, 2022 at 14:45