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Haskell, 2221 bytes

scanl(+)0$mapf n=sum$map(1/)[1..]n] 

Haskell, 22 bytes

scanl(+)0$map(1/)[1..] 

Haskell, 21 bytes

f n=sum$map(1/)[1..n] 
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Haskell, 22 bytes

scanl(+)0$map(1/)[1..]