Basic Analysis I — Real Analysis · Problem 2.6.4

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · series-rearrangements

Problem

a) Show that the alternating harmonic series

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}

has a rearrangement such that for every interval

(x, y)

, there exists a partial sum

s_n

of the rearranged series such that

s_n \in (x, y)

. b) Show that the rearrangement you found does not converge. See Example 2.6.4. c) Show that for every

x \in \mathbb{R}

, there exists a subsequence of partial sums

\{s_{n_k}\}_{k=1}^{\infty}

of your rearrangement such that

\lim_{k \to \infty} s_{n_k} = x