Question

Let \(t_{n}=\frac{s_{1}+s_{2}+\cdots+s_{n}}{n}, n \geq 1\). (a) Prove: If \(\lim _{n \rightarrow \infty} s_{n}=s\) then \(\lim _{n \rightarrow \infty} t_{n}=s\). (b) Give an example to show that \(\left\\{t_{n}\right\\}\) may converge even though \(\left\\{s_{n}\right\\}\) does not.

Short answer

Question: Prove that if a sequence \((s_n)\) converges to a limit \(s\), then the sequence \((t_n)\), which is the average of the first \(n\) terms of \((s_n)\), also converges to the same limit \(s\). Provide an example of a sequence \((s_n)\) that does not converge, but its corresponding sequence \((t_n)\) does converge. Answer: If a sequence \((s_n)\) converges to a limit \(s\), then using the definition of limits and the properties of limits, we can prove that the sequence \((t_n)\), the average of the first \(n\) terms of \((s_n)\), will also converge to the same limit \(s\). An example of a sequence \((s_n)\) that does not converge but its corresponding sequence \((t_n)\) converges is the sequence \((s_n) = (-1)^n\). This sequence oscillates between -1 and 1 and does not converge. However, its corresponding sequence \((t_n)\) converges to 0.

Step-by-step solution

(a) Proof that if \(\lim _{n \rightarrow \infty} s_{n}=s\) then \(\lim _{n \rightarrow \infty} t_{n}=s\) #

We are given that \(\lim _{n \rightarrow \infty} s_{n}=s\). Let \(\epsilon > 0\) be any positive number. According to the definition of the limit, there exists a positive integer \(N_1\) such that for all \(n \geq N_1\), we have \(|s_n - s| < \frac{\epsilon}{2}\). Now consider the sequence \((t_n)\). We have $$ t_n = \frac{s_1 + s_2 + \cdots + s_n}{n}. $$ For \(n \geq N_1\), we have $$ t_n - s = \frac{s_1 + s_2 + \cdots + s_n}{n} - s = \frac{(s_1 - s) + (s_2 - s) + \cdots + (s_n - s)}{n}. $$ Since \(|s_n - s| < \frac{\epsilon}{2}\) for all \(n \geq N_1\), we get $$ |t_n - s| \leq \frac{|s_1 - s| + \cdots + |s_{N_1 - 1} - s| + \frac{\epsilon}{2} + \cdots + \frac{\epsilon}{2}}{n}. $$ Taking the limit as \(n \rightarrow \infty\), we have $$ \lim_{n \rightarrow \infty} (t_n - s) \leq \lim_{n \rightarrow \infty} \frac{|s_1 - s| + \cdots + |s_{N_1 - 1} - s|}{n} + \frac{\epsilon}{2}, $$ which implies \(\lim_{n \rightarrow \infty} (t_n - s) \leq 0 + \frac{\epsilon}{2}\). By the squeeze theorem, we get \(\lim_{n \rightarrow \infty} (t_n - s) = 0\), which means \(\lim_{n \rightarrow \infty} t_n = s\). Hence, if \(\lim _{n \rightarrow \infty} s_{n}=s\) then $\lim _{n \rightarrow \infty} t_{n}=s$.

(b) An example of a sequence \((s_n)\) that does not converge but \((t_n)\) does converge #

Consider the sequence \((s_n)\) defined as \(s_n = (-1)^n\). This sequence does not converge since it oscillates between -1 and 1. Now, let's compute the sequence \((t_n)\): $$ t_n = \frac{s_1 + s_2 + \cdots + s_n}{n}. $$ For even \(n\), \(t_n = \frac{(-1 + 1) + (-1 + 1) + \cdots + (-1 + 1)}{n} = 0\). For odd \(n\), we have $$ t_n = \frac{(-1 + 1) + (-1 + 1) + \cdots + (-1 + 1) - 1}{n} = \frac{-1}{n}. $$ We can clearly see that \(\lim_{n \rightarrow \infty} t_n = 0\). So, in this case, \((t_n)\) converges to 0 while \((s_n)\) does not converge. This is the required example.

Key concepts

Limit of a Sequence

A sequence is a list of numbers arranged in a specific order. The *limit of a sequence* refers to the value that the terms of the sequence approach as the sequence progresses towards infinity. If the terms of the sequence get arbitrarily close to a number, we say the sequence converges to that limit.

**Understanding Limits**
A sequence \( \{s_n\} \) has a limit \( s \) if for every positive number \( \epsilon \), there is a point beyond which the terms of the sequence stay within \( \epsilon \) of \( s \). Mathematically, we express this as: \( \lim_{n \rightarrow \infty} s_n = s \).

Limits are crucial in understanding the behavior of sequences, especially for realizing how sequences behave over time.

**Applying to Example**
In the provided problem, if the sequence \( \{s_n\} \) converges to \( s \), it has been proven that the arithmetic mean sequence \( \{t_n\} \) also converges to \( s \). This shows the consistency of sequence behavior across different formulations.

Squeeze Theorem

The *squeeze theorem* is a powerful tool in calculus used to find the limit of a sequence or function. It is employed when a sequence is "squeezed" between two other sequences that have the same limit.

**How the Squeeze Theorem Works**
Suppose \( \{a_n\}, \{b_n\}, \{c_n\} \) are sequences such that \( a_n \leq b_n \leq c_n \) for all sufficiently large \( n \). If \( \lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} c_n = L \), then \( \lim_{n \rightarrow \infty} b_n = L \). This means if \( b_n \) is trapped between \( a_n \) and \( c_n \), it must share their limit.

**Application in Problem**
In the original exercise, the squeeze theorem helped show that \( \{t_n\} \) converges to the same limit \( s \) as \( \{s_n\} \). By showing that \( t_n - s \) is squeezed to zero, it's clear that \( \{t_n\} \) must converge to \( s \). The theorem provides a logical pathway to establish convergence confidently.

Arithmetic Mean Sequence

The *arithmetic mean sequence* involves finding the average of terms up to the \( n \)-th term in a sequence. It’s a crucial concept for understanding how averages behave as more data is included.

**Defining the Arithmetic Mean Sequence**
Given a sequence \( \{s_n\} \), the arithmetic mean sequence \( \{t_n\} \) is defined as:

• \( t_n = \frac{s_1 + s_2 + \cdots + s_n}{n} \)

This means \( t_n \) is simply the average of the first \( n \) terms of \( \{s_n\} \).

**Importance and Usage**
This sequence is important because it smoothens out the fluctuations in \( \{s_n\} \). For example, even if \( \{s_n\} \) doesn't converge, like the oscillating sequence \( s_n = (-1)^n \), the arithmetic mean sequence can still converge, as it does to zero in this case. This demonstrates how averaging can lead to a different type of convergence.

**Example from Exercise**
In the given example, even though \( s_n = (-1)^n \) does not converge due to its oscillation, the arithmetic mean sequence \( t_n \) derived from \( s_n \) approaches zero, illustrating a key property of averages in sequences.