If Alex has 4 pens in the first week of a month, she will have 4*2 = <<4*2=8>>8 pens in the second week.
After her pen collection doubles in the third week, Alex will have 2*8 = <<2*8=16>>16 pens.
In the last week of the month, Alex will have 2*16 = <<2*16=32>>32 pens after her pen collection.
If Jane has 16 pens after a month, Alex will have 32-16 = <<32-16=16>>16 more pens than Jane.
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