Understanding alternating series test examples requires familiarity with alternating series, convergence, divergent tests, and estimation of error. An alternating series involves summing alternating positive and negative terms. The convergence test assesses whether a series approaches a finite limit. Divergent tests determine if a series increases or decreases without bound. Estimating the error evaluates the difference between the true value of the series and its approximation. By exploring examples of these concepts, we gain insights into the behavior of alternating series and their applications in mathematics and science.
Structure for Alternating Series Test Examples
The Alternating Series Test is used to determine whether an alternating series converges or diverges.
-
State the alternating series:
- An alternating series is a series whose terms alternate in sign.
- For example, (-1)^n, (-1)^(n+1), (-1)^(n+3) are all alternating series.
-
Check for convergence:
- The Alternating Series Test states that an alternating series “∑ (-1)^n b_n” converges if the following conditions are met:
- b_n > 0 for all n.
- b_{n+1} ≤ b_n for all n.
- lim_(n→∞) b_n = 0.
- The Alternating Series Test states that an alternating series “∑ (-1)^n b_n” converges if the following conditions are met:
-
Evaluate the error:
- If the Alternating Series Test applies, the error in approximating the sum of the series by its first n terms is less than or equal to the absolute value of the (n+1)th term.
- That is, |Error| ≤ |b_{n+1}|.
Example 1:
Series: (-1)^n / n
Check for convergence:
– b_n = 1/n > 0 for all n.
– b_{n+1} = 1/(n+1) < b_n for all n.
- lim_(n→∞) b_n = lim_(n→∞) 1/n = 0.
Conclusion: The series converges by the Alternating Series Test.
| n | Sum | Error |
|---|---|---|
| 10 | 0.4605 | 0.1 |
| 100 | 0.460505 | 0.01 |
| 1000 | 0.460505171 | 0.001 |
Example 2:
Series: (-1)^n / n^2
Check for convergence:
– b_n = 1/n^2 > 0 for all n.
– b_{n+1} = 1/(n+1)^2 < b_n for all n.
- lim_(n→∞) b_n = lim_(n→∞) 1/n^2 = 0.
Conclusion: The series converges by the Alternating Series Test.
| n | Sum | Error |
|---|---|---|
| 10 | 0.3634 | 0.0061 |
| 100 | 0.363397 | 0.000003 |
| 1000 | 0.36339679 | 0.00000001 |
Question 1:
How does the alternating series test determine the convergence of alternating series?
Answer:
The alternating series test states that an alternating series, a series whose terms alternate in sign, converges if the absolute value of the terms decreases, and the limit of the absolute value of the terms approaches zero. In other words, for an alternating series to converge, the terms must decrease in magnitude and eventually become negligible.
Question 2:
What is the relationship between the alternating series test and the absolute series test?
Answer:
The absolute series test is a related test that examines the convergence of a series by considering the absolute value of its terms. If the absolute series test converges, then the alternating series test will also converge. However, the converse is not necessarily true.
Question 3:
Can the alternating series test be applied to series with non-alternating signs?
Answer:
No, the alternating series test is specifically designed for series with alternating signs. If the signs of the terms do not alternate, the test cannot be applied.
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