Let's correct the analysis based on the given series:

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

To determine if this series converges, we need to analyze the behavior of the general term $\frac{2^n}{2^n + 2}$ as $n \to \infty$.

First, let's simplify the general term:

$\frac{2^n}{2^n + 2}$

For large $n$, $2^n$ will dominate over the constant term 2 in the denominator. So, we can approximate the general term as follows:

$\frac{2^n}{2^n + 2} \approx \frac{2^n}{2^n} = 1$

Thus, as $n \to \infty$, the term $\frac{2^n}{2^n + 2}$ approaches 1.

Since the terms of the series do not approach zero as $n \to \infty$, the series diverges. For a series $\sum a_n$ to converge, it is necessary that the terms $a_n \to 0$ as $n \to \infty$. Here, $\frac{2^n}{2^n + 2}$ approaches 1, not 0.

Conclusion

The series $\sum_{n=3}^{\infty} \frac{2^n}{2^n + 2}$ diverges because its general term does not approach zero as $n \to \infty$.

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask next:

1. How do you determine the convergence of a series using the ratio test?
2. Can you explain the comparison test for series convergence?
3. What is the difference between absolute and conditional convergence?
4. How do you determine the radius of convergence for a power series?
5. Can you solve an example using the integral test for series convergence?

Tip: Always check if the terms of a series approach zero before concluding convergence. If they do not, the series cannot converge.