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Chapter 1: Problem 157

Consider the functions \(\varepsilon: \mathbb{Z} \rightarrow\\{1,-1\\}\) having period \(N\), where \(N>1\) is a positive integer. For which periods \(N\) does there exist an infinite series \(\sum_{n=1}^{\infty} a_n\) with the following properties: \(\sum_{n=1}^{\infty} a_n\) diverges, whereas \(\sum_{n=1}^{\infty} \varepsilon(n) a_n\) converges for all nonconstant \(\varepsilon\) (of period \(N)\) ?

Short Answer

Expert verified
The period \(N\) must be 2.

Step by step solution

01

- Understand the Problem

The problem requires us to determine the periods \(N\) for which there exists a series \(\sum_{n=1}^{\infty} a_n\) that diverges, but for which the series \(\sum_{n=1}^{\infty} \varepsilon(n) a_n\) converges for all nonconstant functions \(\varepsilon\) of period \(N\).
02

- Define Function Properties

Identify the properties of the function \(\varepsilon: \mathbb{Z} \to \{1, -1\}\) with period \(N\). For a function to have period \(N\), \(\varepsilon(n + N) = \varepsilon(n)\) for all integers \(n\).
03

- Fourier Series Representation

Recall that any periodic function can be decomposed into a Fourier series. Consider the sequence of partial sums for the series \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} \varepsilon(n) a_n\). Analyze the effect of \(\varepsilon(n)\) on the convergence of the series.
04

- Analyze Contribution of Periodic Function

A periodic function \(\varepsilon(n)\) with period \(N\) can be written as a sum of complex exponentials, \(e^{2 \pi i k n / N}\). The series \(\sum_{n=1}^{\infty} a_n\) is expected to interact with these exponential terms. Assess how this interaction affects convergence.
05

- Identify Suitable Period N

Observe that periodic functions with period \(N\) form a finite-dimensional subspace of \(\ell^\text{鈭瀩\). For \(\sum_{n=1}^{\infty} \varepsilon(n) a_n\) to converge, \(N\) must be such that the series' terms, when modulated by \(\varepsilon(n)\), rapidly decrease. This hints that N has to be 2 for the conditions to hold because modulation by -1 causes alternating series, which is known to converge if the terms decrease to 0.
06

- Finalize the Period N

After checking various cases, the only period that universally satisfies these conditions is \(N = 2\). When \(\varepsilon(n)\) alternates between 1 and -1, the modulated series \(\sum_{n=1}^{\infty} (-1)^n a_n\) converges, even if \(\sum_{n=1}^{\infty} a_n\) diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divergent Series
A series is said to diverge if its sequence of partial sums does not approach a finite limit. Simply put, a divergent series keeps growing without bound or fails to settle at a single value. For example, the series \(\frac{1}{n}\) diverges because its partial sums grow indefinitely. Understanding divergence is crucial since some transformations of a divergent series can still lead to a convergent series. This concept is fundamental in analyzing series involving periodic and alternating functions.
Periodic Functions
A function \(\foo\) is periodic if it repeats its values in regular intervals, called the period. Mathematically, \(\foo(t+N) = \foo(t)\) for all t, where N is the period of the function. Periodic functions are common in nature and mathematics, including waves and orbits. For instance, the sine and cosine functions are periodic with period 2蟺. In our problem context, we consider functions \(\foo: \mathbb{Z} \rightarrow \{1, -1\}\) with period N. These functions play a role in determining convergence behaviors when they modulate series.
Fourier Series
Any periodic function can be expressed as a sum of sine and cosine functions, or more generally, as a sum of complex exponentials. This decomposition is known as a Fourier series. The Fourier series of a function \(f(t)\) with period \(N\) is given by: \[ f(t) = \sum_{k=-\infty}^{\infty} c_k e^{2 \pi i k t / N} \] where the coefficients \(c_k\) are calculated using integrals over one period of the function. Fourier series allow us to analyze the frequency components of periodic signals and are useful in solving problems involving periodic modulation, like the one in the exercise.
Alternating Series
An alternating series is one in which the terms alternate in sign: positive, negative, positive, and so on. An example is the series \(\frac{(-1)^n}{n}\). The alternating series test provides a criterion for convergence: if the terms of the series decrease in absolute value and approach zero, the series converges. In our problem, if a periodic function \(\varepsilon(n)\) has a period of 2, then \(\frac{\varepsilon(n)}{n}\) becomes an alternating series when \(\varepsilon(n) = (-1)^n\). This is why the series \(\frac{(-1)^n a_n}{n}\) converges, even if \(\frac{a_n}{n}\) diverges.
Complex Exponentials
Complex exponentials take the form \(e^{i\theta}\) and are related to trigonometric functions via Euler's formula: \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\). These functions are essential in expressing periodic functions and their Fourier series representations. In our exercise, periodic functions are written as sums of complex exponentials, which simplifies the analysis of their modulation effects. The exponential terms \(e^{2 \pi i k n / N}\) within the Fourier series interact with the series terms \(a_n\), influencing the convergence properties of the modulated series.

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Most popular questions from this chapter

Let \(\alpha=0 . d_1 d_2 d_3 \ldots\) be a decimal representation of a real number between 0 and 1. Let \(r\) be a real number with \(|r|<1\). a. If \(\alpha\) and \(r\) are rational, must \(\sum_{i=1}^{\infty} d_i r^i\) be rational? b. If \(\alpha\) and \(r\) are rational, must \(\sum_{i=1}^{\infty} i d_i r^i\) be rational? c. If \(r\) and \(\sum_{i=1}^{\infty} d_i r^i\) are rational, must \(\alpha\) be rational?

For three points \(P, Q\), and \(R\) in \(\mathbb{R}^3\) (or, more generally, in \(\mathbb{R}^n\) ) we say that \(R\) is between \(P\) and \(Q\) if \(R\) is on the line segment connecting \(P\) and \(Q\) ( \(R=P\) and \(R=Q\) are allowed). A subset \(A\) of \(\mathbb{R}^3\) is called convex if for any two points \(P\) and \(Q\) in \(A\), every point \(R\) which is between \(P\) and \(Q\) is also in \(A\). For instance, an ellipsoid is convex, a banana is not. Now for the problem: Suppose \(A\) and \(B\) are convex subsets of \(\mathbb{R}^3\). Let \(C\) be the set of all points \(R\) for which there are points \(P\) in \(A\) and \(Q\) in \(B\) such that \(R\) lies between \(P\) and \(Q\). Does \(C\) have to be convex?

As is well known, the unit circle has the property that distances along the curve are numerically equal to the difference of the corresponding angles (in radians) at the origin; in fact, this is how angles are often defined. (For example, a quarter of the unit circle has length \(\pi / 2\) and corresponds to an angle \(\pi / 2\) at the origin.) Are there other differentiable curves in the plane with this property? If so, what do they look like?

Suppose we have a configuration (set) of finitely many points in the plane which are not all on the same line. We call a point in the plane a center for the configuration if for every line through that point, there is an equal number of points of the configuration on either side of the line. a. Give a necessary and sufficient condition for a configuration of four points to have a center. b. Is it possible for a finite configuration of points (not all on the same line) to have more than one center?

For what real numbers \(x\) can one say the following? a. For each positive integer \(n\), there exists an integer \(m\) such that $$ \left|x-\frac{m}{n}\right|<\frac{1}{3 n} . $$ b. For each positive integer \(n\), there exists an integer \(m\) such that $$ \left|x-\frac{m}{n}\right| \leq \frac{1}{3 n} $$
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