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 term for sequences whose mean is defined?
The standard term is *Cesàro summable*, named after [Ernesto Cesàro](https://en.wikipedia.org/wiki/Ernesto_Ces%C3%A0ro). 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. 


