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Chapter 3: Problem 11

If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, and if \(b_{n}:=\left(a_{1}+\cdots+a_{n}\right) / n\) for \(n \in \mathbb{N}\), then show that \(\sum b_{n}\) is always divergent.

Short Answer

Expert verified
The series \(\sum b_{n}\) is divergent.

Step by step solution

01

Assumption

Assume that \(\sum b_{n}\) is convergent. This means, there exists a number \(B\), such that for any \(\epsilon > 0\) there exists a number \(N\in \mathbb{N}\) such that if \(n > N\), then \(|b_n - B| < \epsilon\). This condition can be rewritten as \(|nB - \sum_{i=1}^{n}a_i| < n\epsilon \).
02

Bounds for Sequence

Since the sequence \(a_n\) is convergent, for sufficiently large values of \(n\), the terms of the sequence \(a_n\) are close to 0. Thus, there exists an integer \(M\) such that for \(n > M\), \(a_n < \epsilon\). Therefore, for \(n > \max\{N, M\}\), \(n\epsilon > |nB - \sum_{i=1}^{n}a_i| + a_n >= |\frac{n-1}{n}B - \frac{1}{n}\sum_{i=1}^{n-1}a_i| = \epsilon (n-1) / n\). This inequality simplifies to \(n > n - 1\), which is a contradiction because an integer cannot be greater than itself.
03

Conclusion

The contradiction derived in the step 2 shows our initial assumption that \(\sum b_{n}\) is convergent is false. Hence, we conclude that \(\sum b_{n}\) is divergent.

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Most popular questions from this chapter

Establish the convergence and find the limits of the following sequences: (a) \(\left(\left(1+1 / n^{2}\right)^{n^{2}}\right)\). (b) \(\left((1+1 / 2 n)^{n}\right)\), (c) \(\left(\left(1+1 / n^{2}\right)^{2 n^{2}}\right)\). (d) \(\left((1+2 / n)^{n}\right)\).

Let \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) be sequences of positive numbers such that \(\lim \left(x_{n} / y_{n}\right)=0\). (a) Show that if \(\lim \left(x_{n}\right)=+\infty\), then \(\lim \left(y_{n}\right)=+\infty\). (b) Show that if \(\left(y_{n}\right)\) is bounded, then \(\lim \left(x_{n}\right)=0\).

Is the sequence \((n \sin n)\) properly divergent?

Let \(\left(x_{n}\right)\) and \(\left(y_{n}\right)\) be sequences of positive numbers such that \(\lim \left(x_{n} / y_{n}\right)=+\infty\), (a) Show that if \(\lim \left(y_{n}\right)=+\infty\), then \(\lim \left(x_{n}\right)=+\infty\). (b) Show that if \(\left(x_{n}\right)\) is bounded, then \(\lim \left(y_{n}\right)=0\).

Give an cxample of an unbounded sequence that has a convergent subsequence.
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