Recently I came across the nice result that
$$\left\lfloor n \right\rfloor - \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor - \left\lfloor \frac{n}{4}\right\rfloor + \dots \sim n \log 2$$
where $\displaystyle a_n \sim b_n$ means $\displaystyle \lim_{n\to \infty} \frac{a_n}{b_n} = 1$.
So basically, what the above result says is that
$$\left\lfloor n \right\rfloor - \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor - \left\lfloor \frac{n}{4}\right\rfloor + \dots + (-1)^n\left\lfloor \frac{n}{n}\right\rfloor \sim n - \frac{n}{2} + \frac{n}{3} - \frac{n}{4} + \dots + (-1)^n\frac{n}{n}$$
and the fact that we take the integer part has no effect, asymptotically.
So I tried a few other sequences, like the harmonic series, and geometric series with ratio $\frac{1}{2}$, and the results seemed to be similar, but the proofs were dependent on the series in question.
So trying to generalize:
Suppose $a_1, a_2, \dots, a_n, \dots$ is a sequence of non-zero integers, such that $|a_1| \lt |a_2| \lt \dots \lt |a_n| \lt \dots$ and
$$ \sum_{k=1}^{n} \frac{1}{a_k} \sim f(n)$$
(Note that $f(n)$ can be constant).
Let $s_i$ be the sign of $a_i$ (i.e. $s_i = 1$ if $a_i \gt 0$, and $-1$ otherwise).
The question is, is the following true?
$$ \sum_{k=1}^{n} s_k \left\lfloor \frac{n}{|a_k|}\right\rfloor \sim n f(n)$$
I hope it isn't something obvious...