I'm interested in series of the form $\lim_{n\to\infty}$$\sum_{N=1}^n |L-S_N|$ where $\lim_{N\to\infty}S_N = \lim_{N\to\infty}\sum_{k=1}^N a_k = L $ and L is not equal to $0$.
I've tried everything at hand to determine whether or not it always converges. By way of D'Alembert's test I arrive at $$\lim_{N\to\infty}\frac{|L-S_{N+1}|}{|L-S_N|}$$ then making use of the fact that $S_{N+1}=S_N+a_{N+1}$, I get $$\lim_{N\to\infty}|\frac{L-S_N-a_{N+1}}{L-S_N}|$$ $$\lim_{N\to\infty}|1-\frac {a_{N+1}}{L-S_N}|$$ It is the case that $0<S_N<S_{N+1}<L$ then $0<S_{N+1}-S_N<L-S_N$ therefore $0<\frac {a_{N+1}}{L-S_N}<1$ Applying the limit operator on both sides of the inequality $0\leq\lim_{N\to\infty} \frac {a_{N+1}}{L-S_N}\leq 1$ So if the general term divided by the difference between the series and the limit is bigger than $0$ the series will converge, otherwise I don't know. Another way of writting it down would be: $$\lim_{N\to\infty}|1-\frac {a_{N+1}}{a_{N+1} + \sum_{k=N+2}^\infty a_k}|$$ The further the series go, the closer $a_{N+1}$ and $\sum_{k=N+2}^\infty a_k$ they get. Maybe it's $0.5$
I don't know a general result. Here there are some thoughts.
If $a_n>0$ for all $n\geq 1$ then $S_N$ is increasing and $$\sum_{N=1}^{\infty} |L-S_N|=\sum_{N=1}^{\infty} \sum_{k=N+1}^{\infty}a_k=a_2+2a_3+3a_4+\dots=\sum_{k=1}^{\infty}ka_{k+1}.$$ So, for example, if $a_k=1/k^a$, your series is convergent if and only if $a>2$.
This is an example where the sign of $a_k$ is alternating. Let $$S_N=\sum_{k=1}^{N}\frac{(-1)^{k+1}}{k}\to \ln(2)$$ then $$\sum_{N=1}^{2n} |L-S_N|=\sum_{N=1}^{2n}\sum_{k=N+1}^{\infty}\frac{(-1)^{k+N+1}}{k}=\frac{1}{2}\sum_{k=1}^n\frac{1}{k}\to +\infty$$ and your series is divergent.