Alternating Series and Absolute Convergence

When the signs do the work.

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

NameDate

Figure — use these to answer the problems

Alternating signs make the partial sums oscillate, closing on the limit from both sides.

y = 1 sin(0x) + 0

a, amplitude1

Point on the curve1

Figure 1

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
2
What makes a series conditionally convergent?
- It converges, but the series of absolute values diverges.
- Both it and its absolute version converge.
The first omitted term is 0.05. Error bound?
Answer
The first omitted term is 0.001. Error bound?
Answer
How many conditions does the alternating series test require?
Answer

The same ideas with more to keep track of.

3 problems

Does absolute convergence imply convergence? 1 yes, 0 no.
Answer
Does convergence imply absolute convergence? 1 yes, 0 no.
Answer
The absolute version converges too. Absolute or conditional? 1 absolute, 2 conditional.
Answer

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

4 problems

9
Put the alternating series analysis in order.
Write 1 to 4 in the boxes to put these in order.
- Check that the term sizes decrease.
- Check that they approach zero.
- Conclude convergence by the alternating series test.
- Test the absolute version to classify it.
The first omitted term is 0.1. Error bound?
Answer
The Error
The first omitted term of an alternating series is 0.02. What is the error bound?
Answer
The Classification
The alternating harmonic series converges but its absolute version does not. Absolute or conditional? 1 absolute, 2 conditional.
Answer