I don't understand something in your logic/description. It seems to be a minor "typo" type thing, but I'm not sure which statement is correct:

First you said:

"I expected to get 5/2 for n = 2 and 13/3 for n = 3, etc."

But then you said that:

n = 2, sum = 1
n = 3, sum = 5/2
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was correct. Did you want "n=2 to equal 1" or "n=2 to equal 5/2"? (though I can't see how this would mangle the output like it does. I haven't quite figured that part out...*grumble*...:-).

But I sorta mentally ran the 1st loop for n=1. That starts with the inner loop being

   "for my $k (1 .. 0) {"
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which isn't horrible by itself, but certainly isn't what I'd expect for a 1st iteration of your series:

"Sumof(Num/Denom) for Num=(1..n) and Denom='(n..1)'[=>'n+1-Num']"
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