MathDB

decreasing seq, upper bound condition on partial sum, show series upper bound

Source: 2016 South Korea USCM P2

August 16, 2020

real analysisSequenceseriescollege contests

Problem Statement

Suppose

\{a_n\}

is a decreasing sequence of reals and

\lim\limits_{n\to\infty} a_n = 0

. If

S_{2^k} - 2^k a_{2^k} \leq 1

for any positive integer

k

, show that

\sum_{n=1}^{\infty} a_n \leq 1

(At here,

S_m = \sum_{n=1}^m a_n

is a partial sum of

\{a_n\}