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?

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

  • $\begingroup$ Is the "$j$" supposed to be there? $\endgroup$ – HiMatt Oct 6 '19 at 0:14
  • $\begingroup$ @HiMatt: yes, it is just any numeric constant. $c$ was already taken $\endgroup$ – Ross Millikan Oct 6 '19 at 0:18

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