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We are given the series:

\[
\log\left(\frac{1}{2}\right) - \log\left(\frac{2}{3}\right) + \log\left(\frac{3}{4}\right) - \log\left(\frac{4}{5}\right) + \cdots
\]

To test the convergence of this series, we will first simplify the terms and then analyze the behavior of the series.

### Step 1: Simplify the Series
Each term in the series can be written as follows using logarithmic properties:

\[
\log\left(\frac{n}{n+1}\right) = \log(n) - \log(n+1)
\]

The series alternates between positive and negative terms, so the series becomes:

\[
S = \log\left(\frac{1}{2}\right) - \log\left(\frac{2}{3}\right) + \log\left(\frac{3}{4}\right) - \log\left(\frac{4}{5}\right) + \cdots
\]

### Step 2: Look for a Telescoping Pattern
This is a **telescoping series**, where most of the terms cancel out:

\[
S = \left( \log(1) - \log(2) \right) - \left( \log(2) - \log(3) \right) + \left( \log(3) - \log(4) \right) - \cdots
\]

After canceling intermediate terms, we are left with:

\[
S = \lim_{n \to \infty} \left( \log(1) - \log(n+1) \right)
\]

Since \(\log(1) = 0\), the series simplifies to:

\[
S = -\lim_{n \to \infty} \log(n+1)
\]

As \(n \to \infty\), \(\log(n+1) \to \infty\), which means the series diverges to \(-\infty\).

### Conclusion:
The given series **diverges**.

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Would you like further details on any part of this process, or have any questions?

Here are 5 related questions you might explore next:

1. What is a telescoping series, and how is it used to test convergence?
2. Can alternating series converge even if individual terms grow large?
3. What tests other than telescoping series can be used to check convergence?
4. How does the natural logarithm function behave as its argument approaches infinity?
5. Are there conditions under which a series that appears to diverge might actually converge conditionally?

**Tip:** When analyzing series convergence, looking for cancellation patterns can simplify the analysis dramatically, especially with logarithmic or alternating series.

Test the convergence of the series: log(1/2) - log(2/3) + log(3/4) - log(4/5) + ...

Analyze the convergence of a logarithmic series using the telescoping series method. This problem involves understanding logarithmic properties and detecting cancellation patterns for series convergence.

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