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$\sum_{n \in \mathbb{N}} \min(x_n, \frac{1}{n \log n})$ always diverges ?

Source: Simon Marais 2019 A4

October 13, 2019
Sequencesseriescalculusreal analysis

Problem Statement

Suppose x1,x2,x3,…x_1,x_2,x_3,\dotsc is a strictly decreasing sequence of positive real numbers such that the series x1+x2+x3+⋯x_1+x_2+x_3+\cdots diverges.
Is it necessary true that the series ∑n=2∞min⁡{xn,1nlog⁡(n)}\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} } diverges?
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