A counter-example is an example that disproves a statement or theory. It is used to show that a general statement is not always true by providing a specific case where it fails.
The counter-example for $\sum 1/n$ is the Harmonic Series, which is defined as $\sum_{n=1}^{\infty} \frac{1}{n}$. It is divergent, meaning that it does not have a finite sum, despite the fact that each term in the series is positive and tends to zero.
$\sum 1/n$ is a commonly used counter-example because it is a relatively simple series that is easy to understand, but also has important implications in mathematics. It is often used to demonstrate that the convergence of individual terms in a series does not guarantee the convergence of the entire series.
The series $\sum 1/n$ is significant in mathematics because it is an example of a divergent series. This means that it does not have a finite sum, even though each term in the series is positive and tends to zero. It is also used as a counter-example to many theorems and statements in mathematics.
There are some real-life applications of $\sum 1/n$, such as in the study of electrical networks, where it is used to calculate the total resistance of a circuit. It is also used in the analysis of algorithms and in the study of economics and finance. However, in most practical cases, the series is truncated at a certain point to avoid the issue of divergence.