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Chapter 3: Problem 5

Show that the following sequences are not convergent. (a) \(\left(2^{n}\right)\), (b) \(\left((-1)^{n} n^{2}\right)\).

Short Answer

Expert verified
Both sequences (a) \(2^{n}\) and (b) \((-1)^{n} n^{2}\) are not convergent. The first one tends to infinity and the second one oscillates between negative and positive values indefinitely.

Step by step solution

01

Analyzing Sequence (a)

First, consider the sequence \(2^{n}\). This sequence is an exponential sequence, where with each step the previous term is multiplied by 2. Given that the terms increase exponentially, one can infer that this sequence is tending toward infinity rather than a fixed limit.
02

Analyzing Sequence (b)

Next, consider the sequence \((-1)^{n} n^{2}\). This is an alternate sequence where the terms frequently flip between positive and negative values. No matter the value of n, the sign changes with every step. Therefore, no limit could exist, because the value of the sequence does not settle into a fixed value but perpetually oscillates between negative and positive values.
03

Making Conclusions

From the analysis, it's clear that both sequences do not have a fixed limit they are tending to. The first sequence increases indefinitely, so it is tending toward infinity. The second sequence flips between negative and positive values, so it does not settle at a single value. Therefore, both sequences are divergent, which means they are not convergent.

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Most popular questions from this chapter

Show that if \(X\) and \(Y\) are sequences such that \(X\) and \(X+Y\) are convergent, then \(Y\) is convergent.

If \(x_{1}2\), show that \(\left(x_{n}\right)\) is convergent. What is its limit?

Investigate the convergence or the divergence of the following sequences: (a) \(\left(\sqrt{n^{2}+2}\right)\). (b) \(\left(\sqrt{n} /\left(n^{2}+1\right)\right)\). (c) \(\left(\sqrt{n^{2}+1} / \sqrt{n}\right)\), (d) \((\sin \sqrt{n})\).

Let \(\sum a_{n}\) be a given series and let \(\sum b_{n}\) be the series in which the terms are the same and in the same order as in \(\sum a_{n}\) except that the terms for which \(a_{n}=0\) have been omitted. Show that \(\sum a_{n}\) converges to \(A\) if and only if \(\sum b_{n}\) converges to \(A\).

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