A term for sequences whose mean is defined? 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.
 A: 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 $b_i=a_i-a_{i-1}$.  I think Cesàro convergent is probably a better term to use for sequences.   
A: 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. 
