Alternating Series & Types of Convergence

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Exploring series with both positive and negative terms. This stack covers the Alternating Series Test (AST) and the important distinctions between absolute and conditional convergence.

Alternating Series & Types of Convergence

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What is an alternating series?

An alternating series is a series whose terms alternate in sign. It can be written in the form \(\sum (-1)^n b_n\) or \(\sum (-1)^{n+1} b_n\), where \(b_n > 0\).

State the two conditions for the Alternating Series Test (AST) for a series \(\sum (-1)^n b_n\).

The series converges if both conditions are met: 1. The limit of the terms is zero: \(\lim_{n \to \infty} b_n = 0\). 2. The terms are decreasing: \(b_{n+1} \le b_n\) for all \(n\) (or for all \(n\) past a certain point).

What happens if \(\lim_{n \to \infty} b_n \neq 0\) for an alternating series?

The series diverges by the n-th Term Test for Divergence, as the limit of the overall term \(a_n = (-1)^n b_n\) will not be zero.

Does the alternating harmonic series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) converge or diverge?

It converges by the Alternating Series Test. The terms \(b_n = \frac{1}{n}\) approach \(0\) and are decreasing. (\(\lim_{n \to \infty} \frac{1}{n} = 0\) and \(\frac{1}{n+1} < \frac{1}{n}\)).

What is the Alternating Series Remainder (Error Bound)?

For a convergent alternating series whose sum is \(S\), the error in approximating \(S\) by the N-th partial sum \(S_N\) is less than or equal to the absolute value of the first neglected term, \(b_{N+1}\). Formally: \(|R_N| = |S - S_N| \le b_{N+1}\).

What does it mean for a series \(\sum a_n\) to be 'absolutely convergent'?

A series \(\sum a_n\) is absolutely convergent if the series of its absolute values, \(\sum |a_n|\), converges.

What does it mean for a series \(\sum a_n\) to be 'conditionally convergent'?

A series is conditionally convergent if the series itself, \(\sum a_n\), converges, but the series of its absolute values, \(\sum |a_n|\), diverges.

If a series converges absolutely, must it also converge?

Yes. Absolute convergence is a stronger condition. Any series that converges absolutely is guaranteed to converge.

What is the three-step procedure to determine if a series is absolutely convergent, conditionally convergent, or divergent?

1. Test the series of absolute values, \(\sum |a_n|\), for convergence (using Ratio, Root, Integral, etc.). If it converges, the original series converges absolutely. 2. If \(\sum |a_n|\) diverges, check if the original series \(\sum a_n\) is alternating and apply the AST. 3. If AST shows convergence, the series is conditionally convergent. If AST shows divergence, the series diverges.

Is the series \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}\) absolutely convergent, conditionally convergent, or divergent?

1. Test \(\sum |\frac{(-1)^n}{n^2}| = \sum \frac{1}{n^2}\). This is a convergent p-series (\(p=2 > 1\)). 2. Since the series of absolute values converges, the original series is absolutely convergent.

Is the alternating harmonic series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) absolutely or conditionally convergent?

1. Test \(\sum |\frac{(-1)^{n+1}}{n}| = \sum \frac{1}{n}\). This is the harmonic series, which diverges. 2. The original series converges by the AST. 3. Since the original series converges but the absolute series diverges, it is conditionally convergent.

Which convergence tests (Ratio, Root, Integral, Comparison) inherently test for absolute convergence?

All of them. Because these tests require positive terms, you must apply them to the series of absolute values, \(\sum |a_n|\). Therefore, a conclusion of 'convergence' from these tests means 'absolute convergence'.

Alternating Series & Types of Convergence

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