Exact Partial Sum of Harmonic Series
Python 2, 80 79 bytes



D=1;N=n=0;exec"n+=1;y=N=N*n+D;x=D=D*n;"*input()
while y:x,y=y,x%y
print N/x,D/x


Try it online!

Prints the numerator and denominator.

Yay! MathJax support!!!! One observes:

$$\sum_{i=1}^{n} \frac{1}{i} = \sum_{i=1}^{n} \frac{n!}{n!i} = \frac{\sum_{i=1}^{n} \frac{n!}{i}}{n!}$$

Then, thinking about recursion, for \$n\$ positive, in the Numerator:

$$\sum_{i=1}^{n+1} \frac{(n+1)!}{i} = (n+1) \sum_{i=1}^{n} \frac{n!}{i} + \frac{(n+1)!}{n+1} = (n+1) \sum_{i=1}^{n} \frac{n!}{i} + n!$$

and one can't help think of the Denominator \$n!\$ recursively as well; thus the exec.

We must pay the Reduced Fraction Piper with a GCD calculation in the while loop; and then we're done.