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#Mathematica, 21 20 16 bytes
This solution is 1-indexed.
Sum[1./i,{i,#}]&
#Mathematica, 21 2020 16 bytes
Sum[1./i,{i,#}]~N~10&]&
#Mathematica, 21 20 bytes
Sum[1/i,{i,#}]~N~10&
