The answer you accepted is accurate, but more can be said. In short $(C,10)$ is not the limit but it could be called something like the (tenth order) Cesaro limit. And the series is not summable (in the usual sense) and does not have a (usual) limit. But it is Cesaro summable (of order $10$ if you wish.)
First let's move from series to sequences. From a series $ \sum a_i$ We have a a related series $s_1,S_2,\cdots$ of partial sums. If this sequence converges to a value $v$ according to the usual definition we say that the series is a summable series or convergent series. That value is the sum or limit and we might say $v=(C,0). $ Otherwise the series is divergent.
So by definition a divergent series does not have a limit. There are good reasons for those conventions/definitions. But we can also define the Cesaro transform which converts $s_1,s_2,\cdots$ series to $\sigma_1,\sigma_2,\cdots$ and if this new sequence has a limit we say that the original series is Cesaro-summable with Cesaro sum $(C,1).$ (I'm sticking to your notation, although I am not fond of it.)
Note that $(C,2)$ would not be the Cesaro sum of the Cesaro sum but rather the limit of the Cesaro transform of the Cesaro transform of the sequence $s_1,s_2,\cdots.$ We could apply the Cesaro transform to sequences which arose in other ways.
If a series is summable it is also Cesaro summable and $(C,1)=(C,0)$. Sometimes when the convergence to $(C,0)=v$ is slow, the convergence of the Cesaro transform to $C(,1)=v$ is much more rapid.
The whole subject is undeniable cool. How useful it is, I don't know.