State the Alternating Series Test.
If $a_n > 0$, $\lim_{n \to \infty} a_n = 0$, and $a_n$ is a decreasing sequence, then the alternating series $\sum_{n=1}^{\infty} (-1)^n a_n$ converges.
Explain the Alternating Series Test.
If $\lim_{n \to \infty} a_n = 0$ and $a_n$ is decreasing, then the alternating series $\sum (-1)^n a_n$ converges.
Why is it important that $a_n$ decreases in the Alternating Series Test?
It ensures that the terms are getting smaller in magnitude, allowing the partial sums to converge.
What happens if $\lim_{n \to \infty} a_n \neq 0$ in an alternating series?
The series diverges by the Divergence Test.
Does the Alternating Series Test determine absolute convergence?
No, it only determines conditional convergence. It doesn't tell us if $\sum |a_n|$ converges.
What is the significance of $\cos(n\pi)$ in alternating series?
$\cos(n\pi)$ is equivalent to $(-1)^n$, providing the alternating sign for the series.
What is the general form of an alternating series?
$\sum_{n=1}^{\infty} (-1)^n a_n$ or $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$, where $a_n > 0$ for all $n$.
State the first condition for the Alternating Series Test.
$\lim_{n \to \infty} a_n = 0$
State the second condition for the Alternating Series Test.
$a_n$ is a decreasing sequence, i.e., $a_n > a_{n+1}$ for all $n$ beyond some index.
