Specific example of infinite series that converges to $0$

Question

I am looking for an infinite series that all its terms must be distinct integers so that it converges to $0$.

Without the condition specified, there are a lot of examples like
$0+0+0+\dots$
$1+(-1)+0+0+\dots$
$(-2)+1+\frac{1}{2}+\frac{1}{4}+\dots$

If the two conditions must be satisfied, the example I can think of is $$1+(-1)+2+(-2)+3+(-3)+\dots$$ But I do not think this series is a convergent series.

Answer 1

No such series exists. If $\sum_{n=0}^\infty a_n$ converges, then $\lim_{n\to\infty}a_n=0$. So, if each $a_n$ is an integer, $a_n=0$, if $n$ is large enough.

Answer 2

No such series can exist. If an infinite series converges, its terms must become arbitrarily small in absolute value. If those terms are restricted to integers, because there is no "arbitrarily small" integer, there must be some point where all terms from then on are zero. But this then violates the distinctness requirement.