To determine whether the sequence $a_n = \frac{5^n}{5^n + 2}$ converges or diverges, and if it converges, to find its limit, we need to examine the behavior of $a_n$ as $n \to \infty$.

Step-by-Step Solution:

1. Rewrite the Sequence: $a_n = \frac{5^n}{5^n + 2}$

2. Analyze the Limit as $n \to \infty$: To find the limit, consider the dominant terms in the numerator and denominator. As $n \to \infty$, the term $5^n$ in both the numerator and the denominator will dominate over the constant $2$ in the denominator.

3. Divide the Numerator and Denominator by $5^n$: $a_n = \frac{5^n}{5^n + 2} = \frac{\frac{5^n}{5^n}}{\frac{5^n}{5^n} + \frac{2}{5^n}} = \frac{1}{1 + \frac{2}{5^n}}$

4. Take the Limit: As $n \to \infty$, $\frac{2}{5^n} \to 0$ because $5^n$ grows exponentially. Thus: $\lim_{n \to \infty} a_n = \frac{1}{1 + 0} = 1$

Conclusion:

The sequence $a_n = \frac{5^n}{5^n + 2}$ converges, and its limit is: $\lim_{n \to \infty} a_n = 1$

Would you like additional details on this solution?

Here are some follow-up questions related to this problem:

1. How does the behavior of exponential functions help in finding limits of sequences?
2. What happens if the base of the exponential term changes in the denominator?
3. Can we apply a similar approach for sequences involving terms like $n^k$ (polynomials) instead of exponentials?
4. What are some other common methods to determine the convergence or divergence of a sequence?
5. How would this sequence behave if we changed $2$ to a different constant?

Tip: When analyzing the limit of sequences with exponential terms, focus on the dominant terms as $n$ grows large.