When does the sum of a convergent sequence minus its limit converge?

Let $$\{a_n\}$$ be a sequence that converges to $$L$$ and let $$\{b_n\}$$ be a sequence defined as $$b_n := \sum_{k=1}^{n}(a_k-L).$$ Perhaps this is a vague question, but what restrictions do we have to place on $$\{a_n\}$$ in order for $$\{b_n\}$$ to converge? Or, more to the point, what are some nontrivial (i.e. $$\sum_{k=1}^{\infty}a_k$$ coverges) conditions that $$\{a_n\}$$ can satisfy so that $$\{b_n\}$$ converges?

• This depends on how fast $a_n$ converges to $L$. – Kavi Rama Murthy Oct 6 '19 at 0:06
• A "convergent sequence minus its limit" is just a "sequence converging to zero". The question is too broad. – metamorphy Oct 6 '19 at 2:22

Define $$c_n=a_n-L$$. Now $$b_n=\sum_{k=1}^nc_k$$ and your usual rules for a series converging apply. If $$|c_n| \lt \frac j{n^{1+\epsilon}}$$ for $$\epsilon \gt 0$$ it converges. The alternating series theorem applies, and so on.
• Is the "$j$" supposed to be there? – HiMatt Oct 6 '19 at 0:14
• @HiMatt: yes, it is just any numeric constant. $c$ was already taken – Ross Millikan Oct 6 '19 at 0:18