Partial Sum of Harmonic Sequence!

Python 3, 27 bytes

h=lambda n:n and 1/n+h(n-1)

JavaScript, 19 18 bytes

1 byte saved thanks to @RickHitchcock

f=a=>a&&1/a+f(--a)

This is 1-indexed.

f=a=>a&&1/a+f(--a)

for(i=0;++i<10;)console.log(f(i))

Mathematica, 21 20 16 bytes

This solution is 1-indexed.

Sum[1./i,{i,#}]&