Is there an example so that the Cesaro mean of $\{a_n\}$ converges to zero, but $\{a_n\}$ isn’t? (where $a_n>0$)
In case of $a_n\ge0$, there is a well-known example.
But, I couldn’t find an example in case of $a_n>0$.
Give some advice or hint. Thank you!
Take your favourite example with $a_n\geq 0$, and to each $a_n$ add $2^{-n}$. Now all terms are strictly positive, and we haven't changed any of the relevant convergence properties.