Basic Analysis I — Real Analysis · Problem 2.6.11

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · cesaro-summability

Problem

(Challenging): a) If ∑n=1∞an\sum_{n=1}^{\infty} a_n is convergent to aa (in the usual sense), show that ∑n=1∞an\sum_{n=1}^{\infty} a_n is Ces\`aro summable (see above) to aa. b) Show that in the sense of Ces\`aro ∑n=1∞(−1)n\sum_{n=1}^{\infty} (-1)^n is summable to 1/21/2. c) Let an:=ka_n := k when n=k3n = k^3 for some k∈Nk \in \mathbb{N}, an:=−ka_n := -k when n=k3+1n = k^3 + 1 for some k∈Nk \in \mathbb{N}, otherwise let an:=0a_n := 0. Show that ∑n=1∞an\sum_{n=1}^{\infty} a_n diverges in the usual sense (in fact, both the sequence of terms and the partial sums are unbounded), but it is Ces\`aro summable to 00 (seems a little paradoxical at first sight).

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