Pensieve header: A convergent but absolutely divergent series can be summed to whatever we wish.

a[n_] := ((((-1))^n)/(Log[n+2]))

Manipulate[
s=0; k0=0; k1=0;
ArrayPlot[Partition[Table[
If[s<w,
++k0; s+=a[2k0]; 0,
++k1; s+=a[2k1-1]; 1
],
{1024}
], 32]],
{w,-5,5}
]

DynamicModule[{w=-2.78`},s=0;k0=0;k1=0;ArrayPlot[Partition[Table[If[s<w,++k0;s+=a[2 k0];0,++k1;s+=a[2 k1-1];1],{1024}],32]]]

Ceiling[16/3]