Problem 74
Question
Suppose that \(\sum a_{n}\) is convergent and \(\sum b_{n}\) is divergent. Prove that \(\Sigma\left(a_{n}+b_{n}\right)\) is divergent. Hint: Prove by contradiction, using Theorem \(4 .\)
Step-by-Step Solution
Verified Answer
We assumed that the series \(\Sigma\left(a_{n}+b_{n}\right)\) is convergent. Using Theorem 4, we deduced that if this series is convergent and \(\sum a_{n}\) is convergent, then the series \(\Sigma b_{n}\) must also be convergent. However, this contradicts the given condition that \(\Sigma b_{n}\) is divergent. Hence, our assumption is false, and we can conclude that \(\Sigma\left(a_{n}+b_{n}\right)\) must be divergent.
1Step 1: Hypothesis
Let's assume that the series \(\Sigma\left(a_{n}+b_{n}\right)\) is convergent. This means that there exists a limit for the sum of both series as n approaches infinity.
2Step 2: Use Theorem 4
According to Theorem 4, if the series \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are both convergent, their sum, \(\sum (a_{n} + b_{n})\), must also be convergent. Since we're given that \(\sum a_{n}\) is convergent, we can deduce that if the series \(\Sigma\left(a_{n}+b_{n}\right)\) is convergent, then the series \(\Sigma b_{n}\) must also be convergent.
3Step 3: Contradiction
However, this contradicts our initial conditions. We were given that the series \(\Sigma b_{n}\) is divergent, but we have deduced that it must be convergent if the series \(\Sigma\left(a_{n}+b_{n}\right)\) is convergent. This contradiction proves that our initial assumption (that the series \(\Sigma\left(a_{n}+b_{n}\right)\) is convergent) is false.
4Step 4: Conclusion
Since our assumption that the series \(\Sigma\left(a_{n}+b_{n}\right)\) is convergent led to a contradiction, we can conclude that the series \(\Sigma\left(a_{n}+b_{n}\right)\) is indeed divergent. This completes the proof.
Key Concepts
Convergent SeriesDivergent SeriesProof by ContradictionTheorems in Calculus
Convergent Series
A convergent series is one where the sum of its terms approaches a specific value as you add more terms. Imagine you're adding numbers, and as you keep going, you find that the sum gets closer and closer to a particular number. That's convergence.
The series \( \sum a_{n} \) is an example of a convergent series. This means that, no matter how many terms you add, the total sum will settle at a fixed value.
To determine convergence, we often use tests like the comparison test or the ratio test. These tools help check if a series will stabilize at some point.
The series \( \sum a_{n} \) is an example of a convergent series. This means that, no matter how many terms you add, the total sum will settle at a fixed value.
To determine convergence, we often use tests like the comparison test or the ratio test. These tools help check if a series will stabilize at some point.
- Comparison Test: Compare with a known convergent series.
- Ratio Test: Use when terms of the series approach zero rapidly.
Divergent Series
A divergent series is the opposite of a convergent one. Here, the sum of its terms does not settle to a specific number. Instead, it could grow indefinitely or not approach any value at all.
In the exercise, \( \sum b_{n} \) is a divergent series. This indicates that no matter how many terms you add, the sum never reaches a stable value.
Understanding divergence is critical because it affects how series combine. When a divergent series is added to a convergent one, like in the hint of our problem, it impacts the overall convergence of the sum.
In the exercise, \( \sum b_{n} \) is a divergent series. This indicates that no matter how many terms you add, the sum never reaches a stable value.
Understanding divergence is critical because it affects how series combine. When a divergent series is added to a convergent one, like in the hint of our problem, it impacts the overall convergence of the sum.
Proof by Contradiction
Proof by contradiction is a logical method used to show something must be true by proving that assuming the opposite led to a contradiction.
In the exercise, we assumed that the series \( \Sigma(a_{n} + b_{n}) \) was convergent. This assumption means that if \( \sum a_{n} \) is convergent, then \( \sum b_{n} \) must also be convergent by Theorem 4.
However, this conclusion contradicts the initial fact: \( \sum b_{n} \) is actually divergent. Hence, our initial assumption must be wrong—a classic contradiction. If assuming the series is convergent led to a false statement, it shows the series must be divergent instead.
In the exercise, we assumed that the series \( \Sigma(a_{n} + b_{n}) \) was convergent. This assumption means that if \( \sum a_{n} \) is convergent, then \( \sum b_{n} \) must also be convergent by Theorem 4.
However, this conclusion contradicts the initial fact: \( \sum b_{n} \) is actually divergent. Hence, our initial assumption must be wrong—a classic contradiction. If assuming the series is convergent led to a false statement, it shows the series must be divergent instead.
Theorems in Calculus
Theorems in calculus are fundamental truths that help us solve problems related to limits, integrals, and series. In this context, Theorem 4 plays a crucial role.
Theorem 4 states that if two series are both convergent, then their sum is convergent. It's a straightforward principle that helps in problems dealing with multiple series.
Theorem 4 states that if two series are both convergent, then their sum is convergent. It's a straightforward principle that helps in problems dealing with multiple series.
- If \( \sum a_{n} \) and \( \sum b_{n} \) are both convergent, then \( \sum (a_{n} + b_{n}) \) is convergent.
- In our exercise, Theorem 4 aids in understanding why assuming \( \Sigma(a_{n} + b_{n}) \) is convergent led to a contradiction, because one of the series was divergent.
Other exercises in this chapter
Problem 73
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