Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 1 / 3F / Solution

The series ${\sum }_{n\ge 1}{a}_n$ converges to $A$ when its partial sums $s_n$ have limit of a sequence $A$: for every $\epsilon >0$, there is $N$ such that $|s_n-A|<\epsilon$ for all $n\ge N$.

Its averages are the Cesàro means of the sequence of partial sums. To prove their convergence, choose $N$ with $|s_k-A|<\epsilon /2$ for $k\ge N$. ThenThe finite first sum divided by $n$ tends to zero. This proves the Cesàro theorem for convergent sequences in this setting, andThe converse fails. Take $a_n=(-1{)}^{n+1}$, so $s_n=1$ for odd $n$ and $s_n=0$ for even $n$. The partial sums do not converge, butThus the alternating series is Cesàro summable without having an ordinary sum.