MathDB

find the maximum value of t so that b_n becomes unbounded

Source: STEMS 2024, CAT B - P3, CAT A - P5

December 17, 2023

algebra

Problem Statement

Let

r

s

be real numbers, find maximum

t

so that if

a_1, a_2, \ldots

is a sequence of positive real numbers satisfying

a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t

for all

n \ge 2023

then the sum

b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s}

is unbounded, i.e for all positive reals

M

there is an

n

such that

b_n > M