Question

Suppose that \(\left\\{a_{n}\right\\}_{1}^{\infty}\) is monotonic and \(\lim _{n \rightarrow \infty} a_{n}=0 .\) Show that \(\sum_{n=1}^{\infty} a_{n} \sin n x\) and \(\sum_{n=1}^{\infty} a_{n} \cos n x\) define functions continuous for all \(x \neq 2 k \pi(k=\) integer).

Short answer

Question: Prove that the series \(\sum_{n=1}^{\infty} a_{n} \sin n x\) and \(\sum_{n=1}^{\infty} a_{n} \cos n x\) represent continuous functions for all \(x \neq 2 k \pi\) where k is an integer, given that the monotonic sequence \(\left\\{a_{n}\right\\}_{1}^{\infty}\) converges to 0. Answer: Both the sine and cosine series satisfy the conditions of Dirichlet's Test for Uniform Convergence on any compact set not containing any points of the form \(x = 2k\pi\), hence they are uniformly convergent on this set. Since uniformly convergent series represent continuous functions, the series \(\sum_{n=1}^{\infty} a_{n} \sin n x\) and \(\sum_{n=1}^{\infty} a_{n} \cos n x\) are continuous for all \(x \neq 2 k \pi\).

Step-by-step solution

Use Dirichlet's Test for Uniform Convergence

We will use Dirichlet's Test for Uniform Convergence to prove that the given series are continuous. A real series \(\sum_{n=1}^{\infty} a_{n} f_{n}(x)\) converges uniformly on a set S if it satisfies two conditions: (1) \(\left\{a_{n}\right\}\) converges monotonically to 0, and (2) for each x in S, the sequence \(\left\{F_{n}(x)\right\}\) is bounded, where \(F_{n}(x) = \sum_{k=1}^{n} f_{k}(x)\).

Check conditions for the sine and cosine series

To show that the given series satisfy the conditions of Dirichlet's Test, we need to check the conditions for both sine and cosine series. For the sine series, we have: 1. \(\left\\{a_{n}\right\\}\) monotonically tends to 0 (by given assumption). 2. Let \(\displaystyle F_{n}^{s}(x)=\sum_{k=1}^{n} \sin kx\). Note that \(\displaystyle \sum_{k=1}^{n} \sin kx = \operatorname{Im}\left( \sum_{k=1}^{n} e^{ikx} \right)\). For all x in S, where S is a compact set not containing any points of the form \(2k\pi\), it is possible to find a bounded \(\{F_{n}^{s}(x)\}\). For the cosine series, we follow a similar approach: 1. \(\left\\{a_{n}\right\\}\) monotonically tends to 0 (by given assumption). 2. Let \(F_{n}^{c}(x)=\sum_{k=1}^{n} \cos kx\). Note that \(\displaystyle \sum_{k=1}^{n} \cos kx = \operatorname{Re}\left( \sum_{k=1}^{n} e^{ikx} \right)\). For all x in S, where S is a compact set not containing any points of the form \(2k\pi\), it is possible to find a bounded \(\{F_{n}^{c}(x)\}\).

Prove uniform convergence

As both the sine and cosine series satisfy the conditions of Dirichlet's Test on any compact set not containing any points of the form \(x = 2k\pi\), they are uniformly convergent on such set.

Conclude continuity

When a series is uniformly convergent, it represents a continuous function. Since both the sine and cosine series have been shown to be uniformly convergent on any compact set not containing any points of the form \(x = 2k\pi\), they are continuous on this set, as required.

Key concepts

Uniform Convergence

Understanding uniform convergence is crucial for analyzing the behavior of infinite series within real analysis. It is a stronger form of convergence than pointwise convergence and has significant implications for the continuity of functions. In general terms, a sequence of functions \(f_n(x)\) converges uniformly to a function \(f(x)\) on a set \(S\) if, for every \(\epsilon > 0\), there exists an \(N\) such that for all \(n > N\) and for all \(x\in S\), the absolute difference \( |f_n(x) - f(x)| < \epsilon \) holds. This means that the sequence of functions \(f_n\) gets uniformly close to \(f\) over the entire set \(S\), after some index \(N\), regardless of the specific point \(x\) in \(S\).

For the example of the series \(\sum_{n=1}^{\infty} a_n \sin nx\) and \(\sum_{n=1}^{\infty} a_n \cos nx\), we use Dirichlet's Test to prove uniform convergence. The key idea is to confirm that the sequence of partial sums \(F_n(x)\) is bounded and that the coefficients \(a_n\) form a monotonic sequence that converges to zero. Uniform convergence ensures that the series defines a continuous function, which is particularly important when analyzing series in a mathematical or physical context.

Continuous Functions

In real analysis, continuous functions play a pivotal role. A function is said to be continuous at a point \(x_0\) if the following holds: as \(x\) approaches \(x_0\), the function's value \(f(x)\) approaches \(f(x_0)\). In other words, there should be no jumps, breaks, or holes in the graph of \(f\) at that point. Continuity over an entire interval or domain means that this property holds for every point in that domain.

Given a series that converges uniformly to a function \(f\), this resulting function \(f\) will be continuous as well. This is because uniform convergence ensures that the behavior of the sequence of partial sums closely mirrors that of the limit function \(f\) over the entire domain. In the case of our series with sine and cosine terms, establishing uniform convergence away from points where \(x = 2k\pi\) is vital. At these points, the sine and cosine functions themselves are continuous, thus converting the notion of uniform convergence into a guarantee of continuity for the overall function represented by the series.

Real Analysis

Real analysis is a branch of mathematics that deals with the properties of real numbers, sequences, series, and functions. It provides the foundations for calculus and allows us to rigorously explore limits, continuity, differentiation, integration, and the convergence of sequences and series. The concepts of uniform convergence and continuity are instrumental when working with functions and series in real analysis. The tools and theorems developed in real analysis are applicable to a vast array of subjects in both pure and applied mathematics.

One such tool is Dirichlet's Test for Uniform Convergence, which is a method to determine whether a given series converges and, more importantly, if it converges uniformly. In the exercises at hand, applying real analysis principles helps us understand the behavior of the trigonometric series involved and their implications on continuity, thus bridging the gap between abstract theory and practical applications.

Monotonic Sequence

A monotonic sequence is a sequence of numbers that either never decreases or never increases as it progresses. When we say a sequence is monotonically decreasing, this means that each term is less than or equal to the previous term. Conversely, a monotonically increasing sequence has each term greater than or equal to the preceding term. Monotonicity is a central concept in the analysis of convergence for sequences and series.

In the context of the given series, the monotonic sequence in question is \(\{a_n\}\), which by assumption tends towards zero. This property is a part of the criteria we check when applying Dirichlet's Test. Monotonic behavior, coupled with the limit approaching zero, ensures that certain types of series—such as those with trigonometric functions—will converge. The role that monotonic sequences play in the convergence of series is critical as it provides a clear direction in how the terms of the series behave, which often leads directly to important implications about the convergence and behavior of the series as a whole.