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triple check to make sure you get the answer right

To determine whether the series
\[
\sum_{m=1}^{\infty} \frac{2 \ln(3m)}{m^2}
\]
converges or diverges, we'll use the properties of convergence tests. Let's proceed with the analysis.

### Step 1: Simplify the Expression
The term of the series is:
\[
\frac{2 \ln(3m)}{m^2} = \frac{2 (\ln(3) + \ln(m))}{m^2} = \frac{2 \ln(3)}{m^2} + \frac{2 \ln(m)}{m^2}.
\]

This means the series can be considered as the sum of two series:
\[
\sum_{m=1}^{\infty} \frac{2 \ln(3)}{m^2} + \sum_{m=1}^{\infty} \frac{2 \ln(m)}{m^2}.
\]

### Step 2: Analyzing Each Series
#### 1. Series \(\sum_{m=1}^{\infty} \frac{2 \ln(3)}{m^2}\)
This series is a constant multiple of the convergent \(p\)-series:
\[
\sum_{m=1}^{\infty} \frac{1}{m^2}.
\]
Since \(p > 1\), this series converges.

#### 2. Series \(\sum_{m=1}^{\infty} \frac{2 \ln(m)}{m^2}\)
To analyze this series, we can use the **comparison test** or the **integral test**. We note that as \(m \to \infty\), \(\ln(m)\) grows slower than any polynomial, and this term's decay is faster than the corresponding term in the \(p\)-series with \(p = 2\).

The integral of \(\int_{1}^{\infty} \frac{\ln(x)}{x^2} dx\) converges, confirming that this series also converges.

### Conclusion
Since both individual series converge, the original series \(\sum_{m=1}^{\infty} \frac{2 \ln(3m)}{m^2}\) converges.

### Answer
The series converges.

Would you like more details on any of these steps, or do you have any other questions?

Here are five related questions to expand your understanding:
1. What is the \(p\)-series test, and how does it apply to convergence?
2. How does the comparison test work for determining series convergence?
3. Can you explain the integral test for convergence of series?
4. What are other common tests for series convergence like the ratio test or root test?
5. How does the growth rate of logarithmic functions compare to polynomial functions in series?

**Tip:** When analyzing series, breaking them into simpler known series can help identify convergence or divergence efficiently.

Determine whether the series \( \sum_{m=1}^{\infty} \frac{2 \ln(3m)}{m^2} \) is convergent or divergent.

Learn how to determine if the series \( \sum_{m=1}^{\infty} \frac{2 \ln(3m)}{m^2} \) converges or diverges using key mathematical concepts like the p-series test, integral test, and comparison test. Suitable for college-level mathematics.

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