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.

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.

The standard term is Cesàro summable, named after Ernesto Cesàro. Note that a convergent series is also Cesàro summable, but the converse does not always hold.

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.