Consider the series 1 (a) Does the sequence (a,) = -converge or diverge? (b) Does this series converge or diverge? (c) How many terms are required for the sum to exceed 50?
Consider the series 1 (a) Does the sequence (a,) = -converge or diverge? (b) Does this series converge or diverge? (c) How many terms are required for the sum to exceed 50?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Topic Video
Question
![### Series Analysis
**Consider the series** \( \sum_{n=1}^{\infty} \frac{1}{n} \).
#### Questions:
1. **Does the sequence \( \{a_n\} = \frac{1}{n} \) converge or diverge?**
2. **Does this series converge or diverge?**
3. **How many terms are required for the sum to exceed 50?**
### Detailed Explanation:
For an educational website, here's how each question can be addressed:
#### (a) Sequence Analysis
**Question:**
Does the sequence \( \{a_n\} = \frac{1}{n} \) converge or diverge?
**Explanation:**
A sequence \( \{a_n\} \) converges if the limit \( \lim_{n \to \infty} a_n \) exists and is finite. For the sequence \( \{a_n\} = \frac{1}{n} \):
\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
Since the limit is 0, the sequence \( \{a_n\} \) converges.
#### (b) Series Convergence
**Question:**
Does this series converge or diverge?
**Explanation:**
For the series \( \sum_{n=1}^{\infty} \frac{1}{n} \), we need to determine if the sum of its terms converges to a finite value as \( n \) tends to infinity.
This specific series is known as the harmonic series, which is a well-known example of a divergent series despite its terms tending to zero individually.
Mathematically, this is proven via integral comparison or other methods in analysis, showing that the harmonic series diverges. Therefore:
The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges.
#### (c) Term Calculation for Sum Exceeding 50
**Question:**
How many terms are required for the sum to exceed 50?
**Explanation:**
To find the number of terms \( N \) such that \( \sum_{n=1}^{N} \frac{1}{n} > 50 \):
Recognizing that the harmonic series diverges but slowly, we approximate the partial sum \( S_N \) of the harmonic series using:
\[ S_N](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4f75436-93ee-4e58-a0e3-afe0fa0e1ea5%2Ffb1e8db1-1e48-452a-bc9f-26b20379d502%2Fd16waw4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Series Analysis
**Consider the series** \( \sum_{n=1}^{\infty} \frac{1}{n} \).
#### Questions:
1. **Does the sequence \( \{a_n\} = \frac{1}{n} \) converge or diverge?**
2. **Does this series converge or diverge?**
3. **How many terms are required for the sum to exceed 50?**
### Detailed Explanation:
For an educational website, here's how each question can be addressed:
#### (a) Sequence Analysis
**Question:**
Does the sequence \( \{a_n\} = \frac{1}{n} \) converge or diverge?
**Explanation:**
A sequence \( \{a_n\} \) converges if the limit \( \lim_{n \to \infty} a_n \) exists and is finite. For the sequence \( \{a_n\} = \frac{1}{n} \):
\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
Since the limit is 0, the sequence \( \{a_n\} \) converges.
#### (b) Series Convergence
**Question:**
Does this series converge or diverge?
**Explanation:**
For the series \( \sum_{n=1}^{\infty} \frac{1}{n} \), we need to determine if the sum of its terms converges to a finite value as \( n \) tends to infinity.
This specific series is known as the harmonic series, which is a well-known example of a divergent series despite its terms tending to zero individually.
Mathematically, this is proven via integral comparison or other methods in analysis, showing that the harmonic series diverges. Therefore:
The series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges.
#### (c) Term Calculation for Sum Exceeding 50
**Question:**
How many terms are required for the sum to exceed 50?
**Explanation:**
To find the number of terms \( N \) such that \( \sum_{n=1}^{N} \frac{1}{n} > 50 \):
Recognizing that the harmonic series diverges but slowly, we approximate the partial sum \( S_N \) of the harmonic series using:
\[ S_N
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 5 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning