2
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    ClearAll["Global`*"];    
    f[k_] := (4^k (4^k - 1.) Abs[BernoulliB[2 k]])/(2 k)!;
    data = Table[1/Power[f[n], (2 n - 1)^-1], {n, 1, 300}];
    ListPlot[data, PlotLabel -> $Version]

But a 0 / 1 error warning (Power::infy) was generated.

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  • $\begingroup$ if you replace 4^k - 1. with 4^k - 1 the error message goes away (Version 11.3 Windows 10-64bit). $\endgroup$ – kglr Jan 19 at 7:10
8
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But a 0 / 1 error warning was generated.

The error is 1/0 and not 0/1, it happened because you used machine floating point numbers.

f[k_] := (4^k (4^k - 1.) Abs[BernoulliB[2 k]])/(2 k)!;
n = 89;
1/Power[f[n], (2 n - 1)^-1]

gives

 is too small to represent as a normalized machine number; precision \
 may be lost
 Power::infy

But if you change the function to use exact numbers

f[k_] := (4^k (4^k - 1) Abs[BernoulliB[2 k]])/(2 k)!;
n = 89;
1/Power[f[n], (2 n - 1)^-1]

The error goes away

 ListPlot[data, PlotLabel -> $Version]

Mathematica graphics

Now works

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