Is the average of two convergent series equal to the Cesàro sum of the alternating series?

Question

If we have $(a_n)$ and $(b_n)$ such that $\sum a_n$ converges and $\sum b_n$ converges, I know that we do not necessarily have that $\sum c_n$ (where $(c_n)_n=a_0,b_0,a_1,b_1,\dots$ converges.

But is the Cesàro summation of $\sum c_n$ defined and if so, is it equal to $\frac12(\sum a_n+\sum b_n)$?