This question already has an answer here:
No, there is no nice closed form for the harmonic numbers. There are some very accurate approximations that are easily computed;
$$H_n\approx\ln n+\gamma+\frac1{2n}-\frac1{12n^2}$$
is quite good, where $\gamma\approx 0.5772156649$ is the Euler-Mascheroni constant.