Problem 11 - Analysis with an introduction to proof

Question

Let \(\left(x_{n}\right)\) be a sequence of real numbers and let \(y_{n}=x_{n}-x_{n+1}\) for each \(n \in \mathbb{N}\). (a) Prove that the series \(\sum_{n=1}^{\infty} y_{n}\) converges iff the sequence \(\left(x_{n}\right)\) converges. (b) If \(\sum_{n=1}^{\infty} y_{n}\) converges, what is the sum?

Step-by-Step Solution

Answer

In summary, the series \(\sum_{n=1}^{\infty} y_{n}\) converges if and only if the sequence \(\left(x_{n}\right)\) converges. If the series converges, the sum of the series is given by \(S = x_{1} - \lim_{n \to \infty} x_n\).

Step 1: Part (a) Convergence of sequence implies convergence of series

Assume that the sequence \(\left(x_{n}\right)\) converges. That is, there exists a finite limit \(L\) such that \(\lim_{n \to \infty} x_n = L\). We have \(y_n = x_n - x_{n+1}\). We want to show that the series \(\sum_{n=1}^{\infty} y_{n}\) converges. Consider the partial sum, \(S_{k}=\sum_{n=1}^{k} y_{n}\). We can write it as: \(S_{k}=\left(x_{1}-x_{2}\right)+\left(x_{2}-x_{3}\right)+ \cdots+\left(x_{k}-x_{k+1}\right)=x_{1}-x_{k+1}\) Now taking the limit as k goes to infinity, we find that \(\lim_{k \to \infty} S_{k} = \lim_{k \to \infty}\left(x_{1}-x_{k+1}\right)= x_1 - L\) So, since \(x_{1}-L\) is finite, the series \(\sum_{n=1}^{\infty} y_{n}\) converges.

Step 2: The convergence of series implies convergence of the sequence

Now, we will prove the other direction, that if the series \(\sum_{n=1}^{\infty} y_{n}\) converges, then the sequence \(\left(x_{n}\right)\) also converges. Suppose that the series \(\sum_{n=1}^{\infty} y_{n}\) converges to a finite value S. We have \(y_n = x_n - x_{n+1}\). Then for any \(n \in \mathbb{N}\), \(x_{n+1}=x_{n}-y_{n}\) Iteratively we can find that: \(x_{n} = x_{1} - \sum_{k=1}^{n-1} y_{k}\) Now, we'll need to take the limit as n approaches infinity: \(\lim_{n \to \infty} x_n = \lim_{n \to \infty}\left(x_{1} - \sum_{k=1}^{n-1} y_{k}\right) = x_1 - S\) Since both \(x_{1}\) and \(S\) are finite, \(\lim_{n \to \infty} x_n\) is finite, which means that the sequence \(\left(x_{n}\right)\) converges. Hence we have shown that the series \(\sum_{n=1}^{\infty} y_{n}\) converges if and only if the sequence \(\left(x_{n}\right)\) converges.

Step 3: Part (b) The sum of the convergent series

Now, suppose that \(\sum_{n=1}^{\infty} y_{n}\) converges. We want to find the sum of the series. Previously, we've shown that if the series converges, the sequence \(\left(x_{n}\right)\) also converges, and \(\lim_{n \to \infty} x_n = x_1 - S\) Since we know the series converges, we can rewrite the sum as: \(S = \sum_{n=1}^{\infty} y_{n} = x_{1}-\lim_{n \to \infty} x_n\) Thus, if the series \(\sum_{n=1}^{\infty} y_{n}\) converges, the sum is given by \(S=x_{1}-\lim_{n \to \infty} x_n\).

Key Concepts

Real Number Sequences

When we talk about real number sequences, we are referring to an ordered list of real numbers. Each number in this list is called a term of the sequence, and we often denote terms with a subscript, like this: \(x_n\) for the n-th term. Sequences can be finite or infinite, but in the context of series convergence, we're typically interested in infinite sequences - those that go on forever.

To understand a sequence fully, we look at its behavior as the terms go on indefinitely. This gives us insight into concepts like convergence - does the sequence tend toward a specific value, or does it keep changing?

An example of a real number sequence is the list of fractions \( \frac{1}{n} \) for \( n = 1, 2, 3, \) and so on. As \(n\) gets larger, the fractions get smaller and approach zero. Zero would be the limit of this particular sequence.

Series Convergence

The concept of series convergence is closely related to real number sequences. While a sequence is just a list of terms, a series is the sum of those terms. Specifically, if we have a sequence \(x_n\), the series formed by these terms is written as \( \( \)sum_{n=1}^{\infty} x_n \( \) \) — the sum of \(x_n\) from \( n=1 \) to infinity.

To determine the convergence of a series, we look at the sums of its partial sequences, called partial sums. If the sequence of partial sums has a limit as the number of terms grows without bound, and that limit is a finite number, we say the series converges to that limit. If the sequence of partial sums increases without bound, or oscillates without approaching a finite limit, the series diverges.

Partial Sums

The notion of partial sums is integral in understanding series convergence. A partial sum is essentially the sum of the first few terms of a series. For example, if we have a series \( \( \)sum_{n=1}^{\infty} x_n \( \) \) and we want to know the sum of the first four terms, then we look at the partial sum \( S_4 = x_1 + x_2 + x_3 + x_4 \).

The series' behavior is revealed as we look at larger and larger partial sums, denoted by \( S_k \) for the first \( k \) terms. If the sequence \( S_k \) has a limit as \( k \) increases to infinity, this is what we consider when determining the convergence of the full series. Understanding partial sums allows us to break down an infinite process into finite pieces we can calculate.

Limits of Sequences

Finally, the concept of limits of sequences is a foundation of calculus and analysis, which comes into play when discussing the convergence of both sequences and series. A limit describes the value that a sequence approaches as its terms progress towards infinity. Mathematically, we express this as \( \( \)lim_{n \to \infty} x_n = L \( \) \) if the values of \(x_n\) get closer and closer to \(L\) as \(n\) becomes very large.

If a sequence has a limit, we say the sequence is convergent; otherwise, it is divergent. In the context of the exercise we are considering, the series \( \( \)sum_{n=1}^{\infty} y_n = x_1 - \( \)lim_{n \to \infty} x_n \( \) \) converges if the limit of the sequence \(x_n\) exists, providing a concrete example of how the limit of a sequence directly affects the convergence of a related series.