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.