Series involving error function In a problem of probability I obtained the following summation
$$
\sum_{n=0}^\infty [ \mathrm{erf} \ (1+n)k - \mathrm{erf} \ nk ]^2
$$
but I have no idea of how to sum it. I observed (through numerical calculation) that the sum is directly proportional to $k$, at last for $k\sim 10^{-3}$.
For large $n$ the value of both error functions is very close to $1$. Then, the 'most important' terms of the sum are the ones of small $n$. Then, I thought, the term corresponding to $n=0$ would give a reasonable approximation to the sum. If $k$ is small, the term for $n=0$ is approximately $k^2$, which is a completely different behavior than the observed. I checked the terms of the summation and many of then are important, therefore we can not obtain the behavior of the summation only from the largest term.
How can I obtain at least the behavior of the sum with $k$, or how to obtain an approximate value for the sum?
 A: I realized a way to calculate asymptoticly the sum for small and large $k$. In the limit $k\to 0$ we have
$$
\lim_{k\to 0} S(k) = \lim_{k\to 0} k\sum_{n=0}^\infty \frac{[\mathrm{erf}(kn+k)-\mathrm{erf}(kn)]^2}{k^2} k = \ k \int_0^\infty \left[\frac{d}{dx} \mathrm{erf} \ x \right]^2 \ dx,
$$
in which $x=kn$. Evaluating the expression in braces leads to
$$
\frac{4k}{\pi} \int_0^\infty \exp (-2 x^2) \ dx,
$$
therefore,
$$
\lim_{k\to 0} S(k) = \sqrt{\frac{2}{\pi}} k
$$
On the other hand, in the limit $k\to \infty$, all the terms are negligible, since they will be approximately $1-1$, except the first one, that will be
$$
\mathrm{erf}(k)-\mathrm{erf}(0) = 1,
$$
then,
$$
\lim_{k\to \infty} S(k) = 1.
$$
The following graph compares the numerical solution (continuous line) and the asymptotic expressions (dashed lines).

A: This is not an answer since only based on numerical simulation.
If we compute $$S(k)=\sum_{n=0}^\infty \left(\text{erf}( (n+1)k)-\text{erf}( nk)\right)^2$$ what can be noticed is that the result is almost linear for the range $0 \leq k \leq 1$ as shown below
$$\left(
\begin{array}{cc}
 k & S(k) \\
 0.0 & 0.000000 \\
 0.1 & 0.079722 \\
 0.2 & 0.159047 \\
 0.3 & 0.237586 \\
 0.4 & 0.314966 \\
 0.5 & 0.390834 \\
 0.6 & 0.464869 \\
 0.7 & 0.536766 \\
 0.8 & 0.606113 \\
 0.9 & 0.672113 \\
 1.0 & 0.733460
\end{array}
\right)$$ For $k >2$, $0.99 < S(k) <1$.
