Is there any example such that $\lim\limits_{k \to \infty} S_{2k} < \infty$ but $\lim\limits_{k \to \infty} S_k = \infty$?

Question

Let $S_k = \sum\limits_{n=1}^{k}a_n$ be a series. Is there any example that $\lim\limits_{k\rightarrow\infty}S_{2k}$ exists but $\sum\limits_{n=1}^{\infty}a_n$ is convergent.

This question makes no sense. Is there a missing "not" before the last word? — David G. Stork, Mar 24 '21 at 17:50
It would also help, once you edit your query, to please make sure that your query's title is kept consistent with your query. — user2661923, Mar 24 '21 at 17:54
@Gary Difference of that series so that $a_n:=0,1,-1,2,-2,3,...$. — Dole, Mar 24 '21 at 19:47

score 1 · Accepted Answer · answered Mar 24 '21 at 18:25

1

Yes. Counter-example is easy to construct.

We try to choose $(a_{n})$ such that $S_{n}=\begin{cases} n, & \mbox{if }n\mbox{ is odd}\\ 0, & \mbox{if }n\mbox{ is even} \end{cases}.$ This is possible by solving equations. For example, $a_{1}=S_{1}=1$, $a_{2}=S_{2}-S_{1}=-1$, $a_{3}=S_{3}-S_{2}=3$, etc...

Clearly $\lim_{n\rightarrow\infty}S_{2n}=0$ while $\lim_{n\rightarrow\infty}S_{n}$ does not exist.

answered Mar 24 '21 at 18:25

Danny Pak-Keung Chan

12,250

Note that it is impossible that $\lim_n S_n = \infty$ because $(S_n)$ has a convergent subsequence. – Danny Pak-Keung Chan Mar 24 '21 at 18:27
we say lim S_n DNE because it alternates right? – Aseon Mar 24 '21 at 19:28

Is there any example such that $\lim\limits_{k \to \infty} S_{2k} < \infty$ but $\lim\limits_{k \to \infty} S_k = \infty$?

1 Answers1