(Challenging)
a) If ∑n=1∞an is convergent to a (in the usual sense), show that ∑n=1∞an is Ces\`{a}ro summable (see above) to a.
b) Show that in the sense of Ces\`{a}ro ∑n=1∞(−1)n is summable to 1/2.
c) Let an:=k when n=k3 for some k∈N, an:=−k when n=k3+1 for some k∈N, otherwise let an:=0. Show that ∑n=1∞an diverges in the usual sense (in fact, both the sequence of terms and the partial sums are unbounded), but it is Ces\`{a}ro summable to 0 (seems a little paradoxical at first sight).