Question

Prove: If \(0 \leq b_{n} \leq a_{n} \leq b_{n+1},\) then \(\sum a_{n}\) and \(\sum b_{n}\) converge or diverge together.

Short answer

Question: Prove that if the given inequality \(0 \leq b_{n} \leq a_{n} \leq b_{n+1}\) holds for positive integer \(n\), then the series \(\sum a_{n}\) and \(\sum b_{n}\) converge or diverge together. Answer: By showing that both partial sums follow the same increasing or decreasing behavior and using the Comparison Test, we proved that if \(\sum b_{n}\) converges(diverges), then \(\sum a_{n}\) also converges(diverges). Therefore, the series \(\sum a_{n}\) and \(\sum b_{n}\) either converge or diverge together.

Step-by-step solution

Write down the given information

We are given that \(0 \leq b_{n} \leq a_{n} \leq b_{n+1}\). We need to prove that the series \(\sum a_{n}\) and \(\sum b_{n}\) converge or diverge together.

Define partial sums

Let's define the partial sums for both series as \(S_a = \sum_{n=1}^{k} a_{n}\) and \(S_b = \sum_{n=1}^{k} b_{n}\).

Use the given inequality to compare the partial sums

Since \(0 \leq b_{n} \leq a_{n} \leq b_{n+1}\), we can write: \(S_b = \sum_{n=1}^{k} b_{n} \leq \sum_{n=1}^{k} a_{n} = S_a\) And likewise, \(S_a = \sum_{n=1}^{k} a_{n} \leq \sum_{n=1}^{k} b_{n+1} = S_b - b_1 + b_{k+1}\)

Use the Comparison Test

Now, let's consider two cases: Case 1: \(\sum b_{n}\) converges. In this case, \(S_b\) is a bounded increasing sequence, implying that \(S_a\) is also a bounded increasing sequence. Therefore, \(\sum a_{n}\) converges by the Monotone Convergence Theorem. Case 2: \(\sum b_{n}\) diverges. In this case, \(S_b\) is an unbounded increasing sequence, implying that \(S_a\) is also an unbounded increasing sequence. Therefore, \(\sum a_{n}\) diverges.

Conclusion

In both cases, we see that if \(\sum b_{n}\) converges(diverges), then \(\sum a_{n}\) also converges(diverges). So, the series \(\sum a_{n}\) and \(\sum b_{n}\) converge or diverge together.

Key concepts

Comparison Test

The Comparison Test is a valuable tool for determining the convergence or divergence of series. It involves comparing the terms of two series to establish their behavior. In the context of this problem, we are given two sequences, \(a_n\) and \(b_n\), with the relationship \(0 \leq b_n \leq a_n \leq b_{n+1}\). This means each term of \(a_n\) is squeezed between \(b_n\) and \(b_{n+1}\).

This comparison helps us determine that if the series \(\sum b_n\) converges, then \(\sum a_n\) must also converge. If \(\sum b_n\) diverges, then \(\sum a_n\) must also diverge.

• If \(b_n\) is small and the series converges, \(a_n\) being larger but still bounded will also converge.
• If \(b_n\) is large enough to diverge, \(a_n\) being between \(b_n\) and \(b_{n+1}\) will show similar divergence.

Utilizing this test effectively involves ensuring the conditions for sequence comparison are met, as shown in the given inequalities.

Monotone Convergence Theorem

The Monotone Convergence Theorem is another essential concept in understanding series convergence. The theorem states that a sequence that is both monotone (either entirely non-increasing or non-decreasing) and bounded will converge to a limit. In our problem, we see this theory at work in the way that \(S_b\), the series of partial sums for \(b_n\), acts.

For Case 1, if \(\sum b_n\) converges, \(S_b\) forms a bounded and increasing sequence. Thus, by the Monotone Convergence Theorem, \(S_a\) will also be bounded and thus converge because it can't exceed \(S_b\).

• This results from the fact that each partial sum included in \(S_a\) is smaller than the corresponding sum in \(S_b\).
• The theorem provides a critical basis for concluding convergence in scenarios where direct calculation or testing isn't straightforward.

Therefore, the theorem helps us prove that \(\sum a_n\) behaves in convergence similarly to \(\sum b_n\) when both conditions are involved.

Partial Sums

Partial sums are the foundation for understanding the sum of an infinite series. When we talk about partial sums, we mean the sums of the first \(k\) terms of the sequence, \(\sum_{n=1}^{k} a_n\) for our sequence \(a_n\), and \(\sum_{n=1}^{k} b_n\) for \(b_n\).

These sums help us see how series behave over increasing numbers of terms and support proving convergence or divergence. They allow us to track how the sum trends as more terms are added. If a partial sum approaches a finite value as \(k\) increases, the series converges. If not, it diverges.

• In the exercise, the definition and manipulation of partial sums (\(S_a\) and \(S_b\)) illustrate how terms relate and lead to a conclusion about the series' nature.
• Understanding how partial sums function is critical as they offer a practical form of the infinity the series is advancing towards.

Thus, partial sums provide a clear insight into how infinite series behave and are pivotal in applying comparison tests and the monotone convergence theorem.