Alternating Series and Absolute Convergence

When the signs do the work.

12 problems · about 22 minutes · LIM-7.A

What this lesson teaches

The student applies the alternating series test, bounds the error, and distinguishes absolute from conditional convergence.

An alternating series converges when its term sizes decrease to zero.
The error is smaller than the first omitted term.
Absolute convergence implies convergence; conditional convergence does not survive rearrangement.

Straightforward practice. Get the method working first.

5 problems

The first omitted term of an alternating series is 0.004. What is the error bound?
Answer 0.004
Why 0.004.
What makes a series conditionally convergent?
Answer It converges, but the series of absolute values diverges.
Why Convergence that depends on the alternating signs.
The first omitted term is 0.05. Error bound?
Answer 0.05
Why That size.
The first omitted term is 0.001. Error bound?
Answer 0.001
Why That size.
How many conditions does the alternating series test require?
Answer 2
Why Decreasing and approaching zero.

The same ideas with more to keep track of.

3 problems

Does absolute convergence imply convergence? 1 yes, 0 no.
Answer 1
Why It is the stronger property.
Does convergence imply absolute convergence? 1 yes, 0 no.
Answer 0
Why The alternating harmonic series.
The absolute version converges too. Absolute or conditional? 1 absolute, 2 conditional.
Answer 1
Why Both converge.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the alternating series analysis in order.
Answer 1. Check that the term sizes decrease. 2. Check that they approach zero. 3. Conclude convergence by the alternating series test. 4. Test the absolute version to classify it.
Why Classification comes after convergence is established.
The first omitted term is 0.1. Error bound?
Answer 0.1
Why That size.
The Error: The first omitted term of an alternating series is 0.02. What is the error bound?
Answer 0.02
Why 0.02.
The Classification: The alternating harmonic series converges but its absolute version does not. Absolute or conditional? 1 absolute, 2 conditional.
Answer 2
Why Conditional.