MHB Is the Harmonic Series a Counterexample to a Convergent Series?

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The harmonic series, represented as $\sum_{n=1}^{\infty}\frac{1}{n}$, serves as a counterexample to the claim that if $na_n \to 0$, $a_n \ge 0$, and $a_n$ is decreasing, then $\sum a_n$ converges. The terms of the harmonic series satisfy the conditions of being positive and decreasing, with $na_n = 1$ approaching 0 as $n$ increases. Despite meeting these criteria, the harmonic series is known to diverge. This demonstrates that the initial statement does not hold universally. Thus, the harmonic series effectively disproves the assertion regarding convergent series.
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Give a counterexample to

$na_n\to 0,\ a_n\ge 0,\ a_n$ decreasing $\implies\sum a_n$ converges.
 
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Consider $a_n=\frac{1}{n\ln n}$.
 
A counterexample to this statement is the harmonic series, $\sum_{n=1}^{\infty}\frac{1}{n}$. This series satisfies the conditions given, as $\frac{n}{n} = 1 \to 0$ as $n\to\infty$, and the terms are all positive and decreasing. However, the harmonic series does not converge, as it is a well-known example of a divergent series. Therefore, the given statement is not always true.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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