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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.

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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.

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.

What is an alternating series?

A series whose terms alternate in sign.

Define convergence in the context of series.

A series converges if the sequence of its partial sums approaches a finite limit.

Define divergence in the context of series.

A series diverges if the sequence of its partial sums does not approach a finite limit.

What is $a_n$ in the context of the Alternating Series Test?

$a_n$ is the non-alternating part of the series, i.e., the terms without the $(-1)^n$ factor.