Basic Analysis I — Real Analysis · Problem 2.6.14

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · power-series

Problem

Suppose

\{c_n\}_{n=1}^{\infty}

is a sequence. Prove that for every

r \in (0, 1)

, there exists a strictly increasing sequence

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

of natural numbers (

n_{k+1} > n_k

) such that

\sum_{k=1}^{\infty} c_k x^{n_k}

converges absolutely for all

x \in [-r, r]