The image shows the following infinite series:

$\sum_{n=1}^{\infty} \frac{3 + 2^n}{4n + 2}$

To solve this, we need to analyze the series term by term.

Analyzing the Series

We can break down the series as:

$\sum_{n=1}^{\infty} \frac{3 + 2^n}{4n + 2} = \sum_{n=1}^{\infty} \left(\frac{3}{4n + 2} + \frac{2^n}{4n + 2}\right)$

This sum is now the sum of two separate series:

$\sum_{n=1}^{\infty} \frac{3}{4n + 2} + \sum_{n=1}^{\infty} \frac{2^n}{4n + 2}$

Simplifying Each Series

1. The First Series

Consider the first series:

$\sum_{n=1}^{\infty} \frac{3}{4n + 2}$

This can be simplified by factoring out the constant:

$\frac{3}{2} \sum_{n=1}^{\infty} \frac{1}{2n + 1}$

The series $\sum_{n=1}^{\infty} \frac{1}{2n + 1}$ is known to diverge because it behaves similarly to the harmonic series, which is known to diverge.

2. The Second Series

Consider the second series:

$\sum_{n=1}^{\infty} \frac{2^n}{4n + 2}$

This is more complex, but it also does not converge due to the rapid growth of $2^n$ compared to the linear growth of $4n + 2$. The series diverges because the terms do not approach zero fast enough as $n$ increases.

Conclusion

Since both individual series do not converge, the original series:

$\sum_{n=1}^{\infty} \frac{3 + 2^n}{4n + 2}$

diverges.

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Would you like to explore more about why the series diverges or see any alternative methods? Here are some related questions you might consider:

1. How can you determine the convergence of a series involving exponential terms?
2. What are some common tests for series convergence?
3. What is the difference between absolute and conditional convergence?
4. How does the comparison test help in determining the convergence of series?
5. What is the role of the ratio test in series convergence analysis?

Tip: When dealing with series, always check the behavior of the terms as $n$ increases to determine convergence or divergence.