Finding A Function For The Harmonic series

Question

Is it possible to formulate a function that can generate the next number in the harmonic series, for instance:

When $$ y = 4,$$ $$x=1+\frac{1}{2} +\frac{1}{3}+ \frac{1}{4}\ldots$$

Thanks for your time and effort.

Answer 1

$$ f(n) = \sum_{j=1}^n \frac{1}{j} $$

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Okay, that's a little facetious. There are other representations and approximations of the harmonic numbers -- you can look through the answers to the question Martin R posted above, and there's a summary on Wikipedia.

But, suppose I rephrase your question like this:

Is there a function which exactly computes the $n^{th}$ harmonic number faster than simply computing the partial sum?

To my knowledge, there is not.