Question

Suppose that \(\lim _{n \rightarrow \infty}\left|s_{n}\right|=\gamma\) (finite). Show that \(\left\\{s_{n}\right\\}\) diverges unless \(\gamma=0\) or the terms in \(\left\\{s_{n}\right\\}\) have the same sign for large \(n .\)

Short answer

Question: Prove that the sequence \(\{s_n\}\) diverges unless \(\gamma=0\) or the terms in \(\{s_n\}\) have the same sign for large \(n\), where \(\lim_{n\rightarrow\infty}|s_n| = \gamma\).

Step-by-step solution

Assumption: \(\gamma > 0\)

We are given that \(\lim_{n\rightarrow\infty}|s_n| = \gamma\) (finite), and let's assume \(\gamma > 0\).

Prove \(\{s_n\}\) diverges if the terms don't have the same sign as \(n \to\infty\)

Suppose the terms in the sequence \(\{s_n\}\) do not have the same sign for large \(n\). Then there must exist two subsequences \(\{s_{n_k}\}\) and \(\{s_{m_k}\}\) such that \(\lim_{k\rightarrow\infty} s_{n_k} > 0\) and \(\lim_{k\rightarrow\infty} s_{m_k} < 0\). Note that either \(\lim_{k\rightarrow\infty} |s_{n_k}| = \gamma\) or \(\lim_{k\rightarrow\infty} |s_{m_k}|= \gamma\). In either case, we have the same results. WLOG, let us assume \(\lim_{k\rightarrow\infty} |s_{n_k}| = \gamma\). Since \(\lim_{n\rightarrow\infty}|s_n| = \gamma\), then \(\lim_{k\rightarrow\infty} |s_{m_k}| = \gamma\) as well. However, \(s_{m_k} < 0\), which implies that the sequence \(\{s_n\}\) cannot converge to a single limit because we have a subsequence that converges to a positive limit and another subsequence that converges to a negative limit. Thus, \(\{s_n\}\) diverges.

When \(\gamma = 0\)

Now let's consider the case when \(\gamma=0\). In this case, we have \(\lim_{n\rightarrow\infty}|s_n| = 0\). By definition of the limit, for a given \(\epsilon>0\), there exists an \(N>0\) such that for all \(n > N\), we have \(|s_n - 0| < \epsilon\), which means \(|s_n| < \epsilon\). This condition does not impose any constraint on the signs of the terms of the sequence. However, since the given sequence converges to \(0\), the sequence \(\{s_n\}\) itself should converge to \(0\) in case the terms have the same sign for large \(n\). If the terms do not have the same sign for large \(n\), the same argument as in Step 1 applies, and we conclude that the sequence diverges. Hence, we have proven that the sequence \(\{s_n\}\) diverges unless \(\gamma=0\) or the terms in \(\{s_n\}\) have the same sign for large \(n\).

Key concepts

Sequence Convergence

In real analysis, the concept of sequence convergence is central to understanding the behavior of sequences. A sequence \( (s_n) \) is said to converge to a limit \( L \) if as \( n \) becomes arbitrarily large, the terms \( s_n \) approach \( L \) ever more closely. Formally, we say that \( (s_n) \) converges to \( L \) if, for every positive number \( \epsilon \) no matter how small, there is a corresponding natural number \( N \) such that for every \( n > N \) it holds that \( |s_n - L| < \epsilon \).

When students encounter convergence, visualizing a sequence's terms getting nearer to a certain value on the number line can be helpful. Such visualization aids in understanding why, for example, \( (s_n) \) with terms \( s_n = 1/n \) converges to \( 0 \): as values of \(\frac{1}{n}\) decrease with increasing \( n \) and get arbitrarily close to \( 0 \) without actually reaching it.

Absolute Value Limit

The absolute value of a sequence plays a pivotal role in determining its behavior. Considering the limit of the absolute value of \( (s_n) \) helps us understand not only the magnitude of the sequence's terms as \( n \) grows but also provides critical insight into the sequence's convergence or divergence.

By definition, \( |s_n| \) is the distance of \( s_n \) from \( 0 \) on the number line. If \( \lim_{n \rightarrow \infty}|s_n| =\gamma \) and \( \gamma \) is a finite number other than \( 0 \), this indicates that the sequence's terms are settling around a distance \( \gamma \) from \( 0 \), but not necessarily converging to a specific point. The step-by-step solution provided in the exercise illustrates that unless \( \gamma=0 \) or the sequence's terms maintain a consistent sign, the sequence \( (s_n) \) will diverge because it fails to satisfy the formal requirement for convergence.

Subsequence Convergence

A subsequence is a sequence derived by selecting terms from an original sequence without changing their order but possibly omitting some. For example, from \( (s_n) \) one might consider a subsequence \( (s_{n_k}) \) where \( k \) indexes the selected terms. Convergence of a subsequence is defined similarly to that of a sequence: \( (s_{n_k}) \) converges to \( L \) if \( |s_{n_k} - L| \) becomes arbitrarily small for sufficiently large \( k \).

Understanding subsequence convergence is vital when considering the overall behavior of sequences. As shown in the exercise solution, if there are two subsequences of \( (s_n) \) that converge to different limits, one positive and one negative, this fact demonstrates that \( (s_n) \) cannot converge to a single limit. Subsequences provide a powerful tool for analyzing sequences by allowing a focus on specific patterns within the broader behavior of \( (s_n) \).

Sequence Divergence Criteria

Determining if a sequence diverges is as important as confirming its convergence. There are several criteria indicating sequence divergence. A sequence \( (s_n) \) is said to diverge if it does not converge to any particular value. This can occur in various ways; the sequence may oscillate between values, grow without bound, or its terms may simply not settle close to any single number.

In the context of the given problem, the sequence \( (s_n) \) diverges if \( \lim_{n \rightarrow \infty}|s_n| =\gamma \) with \( \gamma > 0 \) unless all terms of the sequence have the same sign for sufficiently large \( n \). This stipulation arises because a sequence settling around a nonzero absolute value cannot have terms that both approach and move away from zero—hence it displays divergent behavior. Keeping track of signs and considering absolute value limits are essential when investigating the divergence of a sequence.