An alternating series converges if the term sizes decrease and approach zero. Both conditions are needed.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why it works
The partial sums bounce back and forth by shrinking amounts, closing in on a single value from both sides.
The error bound
The error in stopping after n terms is smaller than the size of the first omitted term. That is a remarkably simple bound.
Absolute convergence
A series converges absolutely if the series of absolute values converges. Absolute convergence implies convergence.
Conditional convergence
A series that converges while its absolute version diverges is conditionally convergent. The alternating harmonic series is the classic case.
Why the distinction matters
A conditionally convergent series can be rearranged to sum to anything at all. Absolute convergence is the stable kind.
The alternating series test
An alternating series converges if the terms decrease in magnitude to zero. Both conditions are needed — decreasing and tending to zero — and both must be stated in a justification.
The error bound
The truncation error is smaller than the first omitted term. That is an unusually simple and strong bound, and the exam asks for it explicitly in series approximation questions.
Absolute against conditional convergence
A series converges absolutely if the series of absolute values converges. If it converges but not absolutely, the convergence is conditional. The alternating harmonic series is the standard conditional example.
Why the distinction matters
A conditionally convergent series can be rearranged to sum to any value at all. Absolute convergence is what makes the sum independent of the order, which is why the categories are separated.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 0)
Step 3: Watch an Example
One step at a time.
Watch Diego Bound an Error
Diego approximates an alternating series by stopping after 4 terms, and the 5th term is 0.02.
- Step 1
The alternating series error bound applies, since the terms decrease to zero.
Step 4: Your Turn
Practice makes it stick.
The Error
Problem 1 of 2
The first omitted term of an alternating series is 0.02. What is the error bound?
The Classification
Problem 2 of 2
The alternating harmonic series converges but its absolute version does not. Absolute or conditional? 1 absolute, 2 conditional.
Signs and Bounds
1 of 8
The first omitted term is 0.05. Error bound?
2 of 8
The first omitted term is 0.001. Error bound?
3 of 8
How many conditions does the alternating series test require?
4 of 8
Does absolute convergence imply convergence? 1 yes, 0 no.
5 of 8
Does convergence imply absolute convergence? 1 yes, 0 no.
6 of 8
The absolute version converges too. Absolute or conditional? 1 absolute, 2 conditional.
7 of 8
Put the alternating series analysis in order.
- 1Check that they approach zero.
- 2Conclude convergence by the alternating series test.
- 3Test the absolute version to classify it.
- 4Check that the term sizes decrease.
8 of 8
The first omitted term is 0.1. Error bound?
Step 5: Quick Check
Show what you know.
Question 1 of 2
The first omitted term of an alternating series is 0.004. What is the error bound?
Question 2 of 2
What makes a series conditionally convergent?
What You Learned
- 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.