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condition for the convergence of sum (x^i/i^b)

Source: Romania National Olympiad 2015, grade xi, p. 3

August 23, 2019

real analysisSequences

Problem Statement

Let be two nonnegative real numbers

a,b

with

b>a,

and a sequence

\left( x_n \right)_{n\ge 1}

of real numbers such that the sequence

\left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1}

is bounded.
Show that the sequence

\left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1}

is convergent.