The image you've uploaded shows the following series:

$\sum_{n=1}^{\infty} \left( (-1)^{n-1} \frac{1}{n} \right)$

This is an alternating series. To analyze its convergence or divergence, we can apply the Cauchy criterion for series, which ensures that a series converges if the partial sums get arbitrarily close as the number of terms increases.

In this case, we have the alternating harmonic series, which can be written as:

$S = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n}$

This series is known to converge by the alternating series test (Leibniz's test), which is a specific form of the Cauchy criterion for alternating series. The test has two conditions:

1. The absolute value of the terms $\frac{1}{n}$ is decreasing.
2. $\lim_{n \to \infty} \frac{1}{n} = 0$.

Both conditions are satisfied, so the series converges.

Would you like to explore more about this convergence or any related concepts?

Here are 5 related questions:

1. What is the sum of the alternating harmonic series?
2. How do we apply the alternating series test (Leibniz’s test) in general?
3. What does the Cauchy criterion state about the convergence of series?
4. How does the alternating harmonic series compare to the regular harmonic series in terms of convergence?
5. Can we use the Cauchy criterion for non-alternating series?

Tip: When testing convergence, always ensure to check the behavior of the terms as $n$ approaches infinity—this is crucial for most convergence tests.