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Explain the Alternating Series Test.

If limnan=0\lim_{n \to \infty} a_n = 0 and ana_n is decreasing, then the alternating series (1)nan\sum (-1)^n a_n converges.

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Explain the Alternating Series Test.

If limnan=0\lim_{n \to \infty} a_n = 0 and ana_n is decreasing, then the alternating series (1)nan\sum (-1)^n a_n converges.

Why is it important that ana_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 limnan0\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 an\sum |a_n| converges.

What is the significance of cos(nπ)\cos(n\pi) in alternating series?

cos(nπ)\cos(n\pi) is equivalent to (1)n(-1)^n, providing the alternating sign for the series.

State the Alternating Series Test.

If an>0a_n > 0, limnan=0\lim_{n \to \infty} a_n = 0, and ana_n is a decreasing sequence, then the alternating series n=1(1)nan\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 ana_n in the context of the Alternating Series Test?

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