A sequence, , ... is Cesàro convergent to if the sequence of its arithmetic
means converges to , i.e.,
Ordinary convergence implies Cesàro convergence to the same limit, but the converse need not hold. For
example, the sequence 1, , 1, , ... does not converge, while its arithmetic
means converge to 0.
, 
, ... is Cesàro convergent to 
if the sequence of its arithmetic
means converges to 
, i.e.,
, 1, 
, ... does not converge, while its arithmetic
means converge to 0.
