The problem involves evaluating or analyzing the infinite series given by:

$\sum_{n=1}^\infty (-1)^n \left( \frac{n+2}{5n+3} \right)^n$

Step-by-Step Solution:

1. Understanding the Series:

   • This is an alternating series due to the term $(-1)^n$, which alternates the sign of each term as $n$ increases.
   • The general term of the series is: $a_n = (-1)^n \left( \frac{n+2}{5n+3} \right)^n$

2. Convergence of the Series: To determine if the series converges, we can use the Alternating Series Test (Leibniz Criterion) and the general test for convergence:

   • For an alternating series, $\sum (-1)^n a_n$ converges if:
     1. $a_n \to 0$ as $n \to \infty$,
     2. $a_n$ is monotonically decreasing (eventually).

3. Limit of $a_n$: Consider the magnitude of the general term: $\left| a_n \right| = \left( \frac{n+2}{5n+3} \right)^n$ As $n \to \infty$, we examine the base $\frac{n+2}{5n+3}$:

   • Divide numerator and denominator by $n$: $\frac{n+2}{5n+3} = \frac{1 + \frac{2}{n}}{5 + \frac{3}{n}} \to \frac{1}{5} \quad \text{as } n \to \infty.$

   Thus, the base of the exponential term tends to $\frac{1}{5}$. Consequently: \left( \frac{n+2}{5n+3} \right)^n \to 0 \quad \text{(since 0 < \frac{1}{5} < 1)}.

   Therefore, $\left| a_n \right| \to 0$.

4. Monotonicity of $a_n$:

   • The term $\left( \frac{n+2}{5n+3} \right)^n$ is positive for all $n \geq 1$.
   • To verify that $\left| a_n \right|$ decreases as $n$ increases, we compare successive terms or use derivatives. This can be done numerically or analytically, but for large $n$, the exponential decay dominates, ensuring monotonicity.

5. Conclusion: Since $\left| a_n \right| \to 0$ and $\left| a_n \right|$ is monotonically decreasing, the series converges by the Alternating Series Test.

Summary:

The given series converges.

Would you like me to:

1. Calculate the approximate sum of the series?
2. Analyze a related series or variant of this one?
3. Discuss convergence tests in more depth?
4. Provide more examples of alternating series?
5. Derive bounds for the sum using the Alternating Series Remainder Theorem?

Tip: For alternating series, always check if the terms go to zero and are decreasing.