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Chapter 22: Problem 5

Prove: if a tail \(\sum_{N}^{\infty} u_{k}(x)\) converges uniformly, then \(\sum_{0}^{\infty} u_{k}(x)\) does also.

Short Answer

Expert verified
If the tail \(\sum_{N}^{\infty} u_{k}(x)\) converges uniformly, then \(\sum_{0}^{\infty} u_{k}(x)\) also converges uniformly.

Step by step solution

01

Understand Uniform Convergence of the Tail

The given is that the tail series \(\sum_{N}^{\infty} u_{k}(x)\) converges uniformly. This means that for any \(\epsilon > 0\), there exists an \(N\) such that for all \(x\) in the domain and for all \(k \geq N\), the inequality \(|u_k(x)| < \epsilon\) holds.
02

Breakdown of the Entire Series

Consider the entire series \(\sum_{0}^{\infty} u_{k}(x)\) which can be written as \(\sum_{0}^{N-1} u_{k}(x) + \sum_{N}^{\infty} u_{k}(x)\). The first part, \(\sum_{0}^{N-1} u_{k}(x)\), is finite and hence convergent by default.
03

Utilize Uniform Convergence of the Tail

Since \(\sum_{N}^{\infty} u_{k}(x)\) converges uniformly, it converges to some function \(S(x)\) uniformly on the domain. This ensures that the partial sums of the tail series, \(s_m(x) = \sum_{N}^{m} u_{k}(x)\), converge to \(S(x)\) and that for \(m\) sufficiently large, \(|s_m(x) - S(x)| < \epsilon\) for all \(x\).
04

Combine Both Parts

The entire series \(\sum_{0}^{\infty} u_{k}(x)\) can be expressed as \(f_N(x) + S(x)\), where \(f_N(x) = \sum_{0}^{N-1} u_{k}(x)\) is finite. Since the finite sum is trivially uniformly convergent and the tail converges uniformly, the entire series must also converge uniformly.

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Key Concepts

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

Tail Series
A "Tail Series" refers to the remaining part of an infinite series after a certain number of initial terms have been summed up. To understand better, consider an infinite series \(危_{0}^{鈭瀩u_{k}(x)\). The "tail" of this series, \(危_{N}^{鈭瀩u_{k}(x)\), represents the sum starting from the N-th term onward.
  • The tail is useful because if this remaining part converges, it often simplifies the analysis of the original series.
  • Understanding the behavior of the tail is crucial in determining the convergence properties of the entire series.
When the "tail" of a series converges uniformly, it implies that beyond some point, the remaining terms collectively approach a limit at the same rate, regardless of the choice of \(x\). This property can have significant implications for determining the uniform convergence of the whole series, as seen in the exercise solution.
Partial Sums
"Partial Sums" are the sums of the first n terms of a series. For example, take the series \(危_{0}^{鈭瀩u_{k}(x)\). The "partial sum" \(s_n(x) = 危_{0}^{n}u_{k}(x)\) represents adding up each term from the first through the n-th.
  • Partial sums can help us understand how a series behaves as more terms are included in the sum.
  • If a sequence of partial sums approaches a specific limit as n increases, the series is said to converge.
In relation to the problem, the concept of partial sums helps break down the series into manageable pieces, essential for analyzing convergence and the effect of uniform properties.
Convergent Series
A "Convergent Series" is one where the sequence of partial sums approaches a finite limit. When we say a series \(危a_k\) converges, it means that as you keep adding new terms indefinitely, the total keeps getting closer to some number.
  • For a series to be convergent, given any small number 蔚, there exists a point in the series beyond which the sum of terms is within 蔚 of this limit.
  • Convergence is essential in calculus and analysis because it helps ensure that infinite processes yield concrete results.
Understanding whether a series is convergent provides insight into its properties and behavior over its domain, aiding in deeper analysis such as uniform convergence discussed later.
Uniform Convergence Definition
"Uniform Convergence" is a stronger form of convergence for sequences of functions. A series of functions \(危u_{k}(x)\) converges uniformly if the speed of convergence does not depend on x.
  • In uniform convergence, for any small number 蔚>0, there exists a threshold N such that for every  x,\(|s_n(x) - S(x)| < 蔚\)
  • Uniform convergence ensures that \(危u_{k}(x)\) behaves smoothly across its domain.
This is significant because, unlike pointwise convergence, uniform convergence preserves the continuity and integrability of the sum of the series if the individual terms possess these properties, making it a crucial concept in analysis and the provided exercise.

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Most popular questions from this chapter

Let \(f_{n}(x)=\sin n x .\) Show that \(f_{n}(x)\) does not converge pointwise on \(\mathbb{R}\).

Let \(f_{n}(x)=\frac{1}{x+n}, x \geq 0, n \geq 1 .\) Find \(f(x)=\lim _{n \rightarrow \infty} f_{n}(x)\), and show the convergence is uniform on \([0, \infty)\).

Prove the following series converge uniformly on the given interval: (a) \(\sum x^{n} \sin n x,[-R, R], 0

Prove that \(\sum_{1}^{\infty} \frac{\cos n x}{n \sqrt{n+1}}\) represents a continuous function for all \(x\).

Let \(f_{n}(x)=e^{-n x} ;\) find \(f(x)=\lim _{n \rightarrow \infty} f_{n}(x)\) on \([0, \infty)\), and show the convergence is uniform on \([R, \infty)\) for any \(R>0\).
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