Problem 13 - Basic Analysis: Introduction to Real Analysis

Question

Let s_ be the kth partial sum of \(\sum x_{n}\). a) Suppose that there exists an \(m \in \mathbb{N}\) such that \(\lim _{k \rightarrow \infty} s_{m k}\) exists and \(\lim x_{n}=0 .\) Show that \(\sum x_{n}\) converges. b) Find an example where \(\lim _{k \rightarrow \infty} s_{2 k}\) exists and \(\lim x_{n} \neq 0\) (and therefore \(\sum x_{n}\) diverges). c) (Challenging) Find an example where \(\lim x_{n}=0,\) and there exists a subsequence \(\left\\{s_{k}\right\\}\) such that \(\lim _{j \rightarrow \infty} s_{k}\) exists, but \(\sum x_{n}\) still diverges.

Step-by-Step Solution

Answer

a) Converges. b) \( x_{n} = (-1)^{n} \). c) \( x_n = \frac{(-1)^n}{\sqrt{n}} \).

Step 1: Understand the Conditions

The problem requires us to show that a series converges if both \( \lim_{k \rightarrow \infty} s_{mk} \) and \( \lim x_{n}=0 \) exist. We need to understand how these conditions affect the properties of convergence.

Step 2: Apply Limit Properties

Since \( \lim_{k \rightarrow \infty} s_{mk} = L \), the partial sum \( s_{mk} \) approaches a limit \( L \). This suggests that the sequence of partial sums is stable and approaches a certain value rather than oscillating or diverging to infinity.

Step 3: Use Convergence Criteria

Given \( \lim x_{n}=0 \) implies the terms in the series become arbitrarily small. The Cauchy criterion for convergence states that for any \( \epsilon > 0 \), there exists \( N \in \mathbb{N} \) such that for all \( n > N \), \( |x_{n}| < \epsilon \). Combined with the stability of partial sums (as \( s_{mk} \) also approaches \( L \)), this implies the series is bounded and converges.

Key Concepts

Partial Sums

In the context of a series, partial sums refer to the sum of several initial terms of a sequence. Imagine you are adding numbers one by one, each time writing down the total. This running total or sum so far is what we call a partial sum.

With the series symbol \( \sum x_n \), the partial sum of the first \( k \) terms is denoted as \( s_k = x_1 + x_2 + \cdots + x_k \). Each \( s_k \) is a specific partial sum, and as \( k \) increases, we get more and more of these sums.

• A key thing to watch is what happens to these partial sums as \( k \) becomes very large.
• There's a special focus on whether the partial sums approach some fixed number, oscillate, or grow without bound.

Understanding partial sums is critical for figuring out whether your original series \( \sum x_n \) converges, meaning everything adds up nicely to a finite number.

Limit of a Sequence

The concept of the limit of a sequence is foundational in understanding series convergence. When we talk about the limit of a sequence, we mean the value that the sequence's terms \( x_n \) get closer and closer to as \( n \) heads towards infinity.

If \( \lim_{n \to \infty} x_n = L \), it means that as \( n \) increases, the numbers in our sequence approach the number \( L \). For series convergence, it's often crucial that \( \lim_{n \to \infty} x_n = 0 \).

• When \( x_n \rightarrow 0 \), it suggests that each new number added to the sequence is getting smaller, potentially giving the series a likelihood of converging.
• If a series has terms not approaching zero, the series cannot converge.

In the exercise, we highlight cases where such limits help us determine convergence and divergence of certain series.

Cauchy Criterion

The Cauchy Criterion is a test for convergence that revolves around the idea of bounding partial sums tightly as you progress along an infinite series.

The criterion states: a series \( \sum x_n \) converges if and only if for every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( m, n > N \), the difference \(|s_m - s_n| < \epsilon\). This means that the terms \( x_n \) can be made arbitrarily small by choosing indexes \( n \) beyond some threshold \( N \).

• This condition essentially tells us that the further we go in adding terms, the total sum doesn't significantly change, indicating stability and potential convergence.
• The importance of using the Cauchy criterion lies in its power to confirm convergence without needing to know the actual limit.

In practical scenarios, like in the exercise, the Cauchy Criterion can simplify verifying series convergence when combined with other limit-based conditions.