To determine whether the given alternating series converges or diverges, we need to analyze the series:

$\sum_{n=1}^{\infty} \frac{(-1)^{n+1} \cdot n}{2n^6 + 1}.$

Step-by-Step Analysis

1. Identify the Alternating Series: The series is of the form $\sum (-1)^{n+1} a_n$ where $a_n = \frac{n}{2n^6 + 1}.$ Since the series has the alternating factor $(-1)^{n+1}$, we can apply the Alternating Series Test (also known as the Leibniz Test).

2. Check Conditions for the Alternating Series Test: The Alternating Series Test requires that:

   • $a_n > 0$,
   • $a_n$ is decreasing, and
   • $\lim_{n \to \infty} a_n = 0$.

   Let's verify each of these conditions:

   • Positivity: Since $n$ and $2n^6 + 1$ are positive for all $n \geq 1$, $a_n > 0$ for all $n$.

   • Monotonicity (Decreasing Behavior): To check if $a_n$ is decreasing, consider that the degree of the denominator ($n^6$) is much higher than that of the numerator ($n$). Thus, as $n$ grows, $a_n$ will decrease because the denominator grows faster than the numerator.

   • Limit Condition: Compute $\lim_{n \to \infty} a_n$: $\lim_{n \to \infty} \frac{n}{2n^6 + 1} = \lim_{n \to \infty} \frac{1}{2n^5 + \frac{1}{n}} = 0.$ Therefore, $\lim_{n \to \infty} a_n = 0$.

   Since all three conditions are satisfied, the Alternating Series Test tells us that the series converges.

Conclusion

The correct answer is:

• C. The series converges by the Alternating Series Test.

Would you like more details, or have any questions on this topic? Here are some related questions:

1. What is the Alternating Series Test, and when is it applicable?
2. How does the rate of growth of terms affect the convergence of a series?
3. Can the Root Test be applied to an alternating series, and under what circumstances?
4. What other tests are commonly used to determine the convergence of series?
5. How does the convergence of a $p$-series compare to that of a geometric series?

Tip: Remember that an alternating series converges if the terms decrease in absolute value to zero, even if the series itself does not converge absolutely.