The function $\frac{2x}{1+x^2}$ produces this curve:
This clearly converges on $0$ across all real numbers. However, summing the function across the half-plane $x\geq 0$ for integer values of $x$ is non-convergent (at least, I think it is) because there is no common ratio.
I'm searching for a similar function in that it is odd, reaches some arbitrary but defined minimum / maximum (depending which half-plane you consider), then decays towards zero. But I am searching for a function where the sum over the half-plane (for integer values of $x$) is convergent.
Can such a curve exist? I don't mind what value it converges on (except for $\infty$), and I don't mind what the minima and maxima are, or where they occur.
Suggestions?
You can get such a function by simply squaring the denominator of the function you gave, i.e. $f(x) = \displaystyle \frac{2x}{(x^2+1)^2}$:
You can see, algebraically and visually, that $f$ remains odd, and when $x>0$, we have:
$$\frac{2x}{(x^2 + 1)^2} < \frac{2x}{x^4} = \frac{2}{x^3}$$
You can prove that $\displaystyle \sum_{n=1}^\infty \frac{2}{n^3}$ converges with, say, the integral test, so $\displaystyle \sum_{n=1}^\infty \frac{2n}{(n^2 + 1)^2}$ converges per the comparison test.