Is the Harmonic Series a Counterexample to a Convergent Series?

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SUMMARY

The harmonic series, represented as $\sum_{n=1}^{\infty}\frac{1}{n}$, serves as a definitive counterexample to the statement that if $na_n \to 0$, $a_n \ge 0$, and $a_n$ is decreasing, then $\sum a_n$ converges. In this case, $a_n = \frac{1}{n \ln n}$ satisfies the conditions of the statement, yet the harmonic series diverges. This demonstrates that the initial conditions do not guarantee convergence, establishing that the statement is not universally valid.

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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.
 

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