Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 2 to ∞)1 / (j ln¹⁰j)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.6.11
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
Verified step by step guidance1
Identify the series given: \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \). This is an alternating series because of the factor \( (-1)^k \), which causes the terms to alternate in sign.
Recall the Alternating Series Test (Leibniz Test), which states that an alternating series \( \sum (-1)^k a_k \) converges if two conditions are met: (1) the sequence \( a_k \) is decreasing, and (2) \( \lim_{k \to \infty} a_k = 0 \).
Check the sequence \( a_k = \frac{1}{2k+1} \). First, verify that \( a_k \) is positive and decreasing. Since the denominator \( 2k+1 \) increases as \( k \) increases, \( a_k \) decreases.
Next, evaluate the limit of \( a_k \) as \( k \to \infty \): \( \lim_{k \to \infty} \frac{1}{2k+1} = 0 \). This satisfies the second condition of the Alternating Series Test.
Since both conditions are satisfied, conclude that the series \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \) converges by the Alternating Series Test.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Alternating Series Test
The Alternating Series Test determines the convergence of series whose terms alternate in sign. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, though not necessarily absolutely.
Recommended video:
10:54
Alternating Series Test
Monotonic Decreasing Sequence
A sequence is monotonic decreasing if each term is less than or equal to the previous term. For the Alternating Series Test, the absolute values of the terms must form such a sequence, ensuring the terms get progressively smaller and approach zero.
Recommended video:
Guided course
8:22Introduction to Sequences
Limit of the Terms Approaching Zero
For a series to converge, the terms must approach zero as the index goes to infinity. This is a necessary condition for convergence; if the terms do not tend to zero, the series diverges regardless of sign alternation.
Recommended video:
05:50One-Sided Limits
Related Practice
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{√(n² + 1) − n}
1
views
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)cos(1 / k⁹)
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{2ⁿ⁺¹3⁻ⁿ}