This may be an extremely stupid and elementary question, but is there a name for sequences $\{a_i\}$ such that $\lim_{n\to\infty} \left( \frac{1}{n}\sum_{i=1}^{n} a_i\right)$ exists? This seems to be a separate condition from boundedness or summability.

6$\begingroup$ Cesàro summable. en.wikipedia.org/wiki/Ces%C3%A0ro_summation $\endgroup$ – Tony Huynh Sep 24 '15 at 20:00

2$\begingroup$ Many thanks. If you want to submit that as an answer I can go ahead and accept it. $\endgroup$ – Tom Solberg Sep 24 '15 at 20:02

$\begingroup$ You're welcome. I made this into an answer as requested. $\endgroup$ – Tony Huynh Sep 24 '15 at 20:11
The standard term is Cesàro summable, named after Ernesto Cesàro. Note that a convergent sequence is also Cesàro summable (with the same limit), but the converse does not always hold.
Edit. I realize that there is some confusion, thanks to the comments of Hurkyl and jeq below. Cesàro summable is usually a property attributed to a series $\sum_i b_i$. We recover the right definition for sequences by regarding the sequence $(a_i)$ as the series $\sum_i b_i$, where $b_1=a_1$ and $b_i=a_ia_{i1}$. I think Cesàro convergent is probably a better term to use for sequences.

$\begingroup$ Perhaps Cesaro mean en.wikipedia.org/wiki/Ces%C3%A0ro_mean? $\endgroup$ – jeq Sep 24 '15 at 20:19

2$\begingroup$ No, the Cesàro means are the sequence of partial averages. A sequence is Cesàro summable if the sequence of Cesàro means converges. $\endgroup$ – Tony Huynh Sep 24 '15 at 20:25

3$\begingroup$ This answer is wrong, according to the wikipedia link. If $a_n$ was the sequence of partial sums of $\sum_k b_k$, then the limit the OP asks about is the Cesaro sum of $\sum_k b_k$. $\endgroup$ – user13113 Sep 24 '15 at 23:47

$\begingroup$ The OP's question, I believe, was for a term for when the limit of a sequence of partial averages exists. The sequence $\lbrace 1,1,...\rbrace$ has a Cesàro mean, but is not Cesàro summable. $\endgroup$ – jeq Sep 25 '15 at 0:00

1$\begingroup$ It is correct. The confusion is the distinction between a series and a sequence. In the wikipedia link provided by @jeq, Cesàro summability is defined as a property of sequences, while in my link above it is defined as a property of series. The property that the OP wants is that the sequence $a_i$ is Cesàro summable, not that the series $\sum a_i$ is Cesàro summable. $\endgroup$ – Tony Huynh Sep 25 '15 at 0:01
The unambiguous name for a sequence with converging Cesàro means of its terms, is Cesàro convergent sequence. The unambiguous name for a sequence with converging Cesàro means of its partial sums, is Cesàro summable sequence.
Now about 'series'. Cauchy (1821, Cours d'Analyse p. 123) described series (French série) as a sequence with real numbers as terms. (In later practice extended to: a sequence with addable (addible?) elements as terms.)
Unfortunately Cauchy proposed and used convergent (French convergente) instead of summable (French sommable) for a sequence/series with converging partial sums. Which resulted  worldwide  in a double meaning for convergent:
 convergent means having a sum in combination with the word series,
 convergent means having a limit in combination with the word sequence.
Conclusion: Cesàro convergent series and Cesàro summable series are synonym. But Cesàro convergent sequence and Cesàro summable sequence are NOT.