The series provided is:

$\sum_{n=0}^{\infty} (-1)^n$

This is an alternating series where each term alternates between $1$ and $-1$.

Let's analyze this:

1. Terms of the series: The terms of the series are: $(-1)^0 = 1, (-1)^1 = -1, (-1)^2 = 1, (-1)^3 = -1, \dots$ So, the series looks like: $1 - 1 + 1 - 1 + 1 - 1 + \cdots$

2. Convergence or divergence:

   • This is known as the alternating harmonic series. However, each term does not decrease in magnitude but remains $1$. This suggests that the partial sums keep oscillating between $1$ and $0$.
   • The partial sums are: $S_1 = 1, \quad S_2 = 1 - 1 = 0, \quad S_3 = 1 - 1 + 1 = 1, \quad S_4 = 1 - 1 + 1 - 1 = 0, \dots$ Therefore, the partial sums do not approach any finite number.

Since the partial sums oscillate between 1 and 0, the series does not converge.

Conclusion:

The series diverges.

Do you have any questions or would you like further details?

Here are 5 related questions to further your understanding:

1. What is the difference between conditional and absolute convergence?
2. Can alternating series sometimes converge? If so, under what conditions?
3. How can you test if a series converges using the Alternating Series Test?
4. What is the behavior of the partial sums of an oscillating series like this one?
5. How does the geometric series relate to this type of problem?

Tip: When dealing with alternating series, it's important to consider the magnitude of terms as well as their sign changes!